Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from...
-
Upload
vincent-munday -
Category
Documents
-
view
225 -
download
4
Transcript of Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from...
![Page 1: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/1.jpg)
Rotational Motion and Angular Momentum
Unit 6
![Page 2: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/2.jpg)
A particle at point P at a fixed
distance r from origin is rotating
about axis O.
Lesson 1 : Angular Position, Velocity, and Acceleration
When a rigid object rotates about its axis, at any given time different parts of the object have different linear velocities and linear
accelerations.
![Page 3: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/3.jpg)
P is at polar coordinate (r, )
is measured counterclockwise from some reference line ( = 0)
s = r
= s
r
![Page 4: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/4.jpg)
is measured in radians (rad)
One radian is the angle subtended by an arc length equal to the radius of the arc.
= s
r
360o = 2r
rrad = 2 rad
1 rad =360o
2= 57.3o
![Page 5: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/5.jpg)
Converting from Degrees to Radians
(rad) =
180o (deg)
90o = /2 rad
60o = /3 rad
270o = 3/2 rad
45o = /4 rad
![Page 6: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/6.jpg)
As particle moves from A to B in time
interval t, the reference line of
length r sweeps out an angle
= f - i
Angular Displacement ()
![Page 7: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/7.jpg)
Average Angular Speed ()
Ratio of the angular displacement of a rigid object to the time interval t.
= f - i
tf - ti
=t
The rad/s is the unit for angular speed.
is positive when increases (counterclockwise motion)
is negative when decreases (clockwise motion)
![Page 8: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/8.jpg)
Instantaneous Angular Speed ()
= limt 0
t
=dt
d
Average Angular Acceleration ()
= f - i
tf - ti
=t
![Page 9: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/9.jpg)
Instantaneous Angular Acceleration ()
= limt 0
t
=dt
d
The rad/s2 is the unit for angular speed.
is positive when object rotates counterclockwise and speeds up
OR
when object rotates clockwise and slows down
![Page 10: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/10.jpg)
Direction of Angular Velocity and Acceleration Vectors
The directions of and are along the axis of rotation.
and are vector quantities with magnitude and direction
![Page 11: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/11.jpg)
Right-Hand Rule
Wrap four fingers of the right hand in
the direction of rotation.
Thumb will point in the direction of angular velocity
vector ().
![Page 12: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/12.jpg)
Example 1
A rigid object is rotating with an angular speed < 0. The angular velocity vector and the angular acceleration vector
are antiparallel. The angular speed of the rigid object is
a) clockwise and increasing
b) clockwise and decreasing
c) counterclockwise and increasing
d) counterclockwise and decreasing
![Page 13: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/13.jpg)
Example 2
During a certain period of time, the angular position of a swinging door is described by = 5.00 + 10.0 t + 2.00 t2, where is in
radians and t is in seconds. Determine the angular position, angular speed, and angular acceleration of the door at
a) at t = 0 b) at t = 3.00 s
![Page 14: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/14.jpg)
Lesson 2 : Rotational Kinematics with Constant Angular Acceleration
=ddt
d = dt
Since =ddt
ddt
i + t =
(by integrating)f = i + t
![Page 15: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/15.jpg)
f = i + it + ½ t2 (by integrating)
Eliminating t from previous two
equations,
f2 = i
2 + 2(f – i)
Eliminating from previous two
equations,
f = i + ½ (i + f) t
![Page 16: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/16.jpg)
xPosition
vVelocity
aAcceleration
![Page 17: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/17.jpg)
Example 1
A wheel rotates with a constant angular acceleration of 3.50 rad/s2.
a) If the angular speed of the wheel is 2.00 rad/s at ti = 0, through what
angular displacement does the wheel rotate in 2.00 s ?
![Page 18: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/18.jpg)
b) Through how many revolutions has the wheel turned during this time interval ?
c) What is the angular speed of the wheel at t = 2.00 s ?
![Page 19: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/19.jpg)
Example 2
A rotating wheel requires 3.00 s to rotate through 37.0 revolutions. Its angular
speed at the end of the 3.00 s interval is 98.0 rad/s. What is the constant angular
acceleration of the wheel ?
![Page 20: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/20.jpg)
Lesson 3 : Angular and Linear Quantities
Every particle of a rigid rotating object moves in a circle whose center is the
axis of rotation.
Tangential speed v = ds
dt
Since s = r
v = ddt
r
v = r
![Page 21: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/21.jpg)
v = r
Tangential Speed
perpendicular distance of a point
from the axis of rotation
angular speed
Tangential speed depends
on distance from axis of
rotation
Angular speed is the same for
all points
![Page 22: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/22.jpg)
Tangential Acceleration
at =dv
dt
Since v = r
at = ddt
r
at = r
![Page 23: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/23.jpg)
Centripetal Acceleration in terms of Angular Speed
ac =v2
r
Since v = r
ac = r2
![Page 24: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/24.jpg)
Total Linear Acceleration
a = at2 + ar
2
a = r22 + r24
a = r 2 + 4
![Page 25: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/25.jpg)
In order to keep the tangential speed of the disc surface at the location of the lens constant, the
disc’s angular speed must vary as the lens moves radially along the disc. In a typical compact disc player, the constant speed of the surface at the
point of the laser-lens system is 1.3 m/s.
Example 1
![Page 26: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/26.jpg)
a) Find the angular speed of the disc in rev/min when information is being read from the innermost first track (r = 23 mm) and the outermost final track (r = 58 mm).
![Page 27: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/27.jpg)
b) The maximum playing time of a standard music CD is 74 min 33 s. How many revolutions
does the disc make during that time ?
![Page 28: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/28.jpg)
c) What total length of track moves past the lens during this time ?
![Page 29: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/29.jpg)
d) What is the angular acceleration of the CD over the 4,473 s time interval ? [Assume that is constant.]
![Page 30: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/30.jpg)
The drive train of a bicycle is shown to the right. The
wheels are 67.3 cm in diameter and pedal cranks 17.5 cm long. The cyclist
pedals at a steady angular rate of 76.0 rev/min. The
chain engages with a front sprocket 15.2 cm in diameter and a rear sprocket 7.00 cm in
diameter.
Example 2
![Page 31: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/31.jpg)
a) Calculate the speed of a link of the chain relative to the bicycle frame.
b) Calculate the angular speed of the bicycle wheels.
![Page 32: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/32.jpg)
c) Calculate the speed of the bicycle relative to the road.
d) What pieces of data, if any, are not necessary for the calculations ?
![Page 33: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/33.jpg)
Lesson 4 : Rotational Kinetic Energy
The total kinetic energy of a rotating rigid object is the sum of the kinetic energies of
its individual particles.
KErot = KEi = ½mivi2
Since v = r,
KErot = ½ miri2i
2
Factoring out 2,
KErot = ½ ( miri2) 2
![Page 34: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/34.jpg)
KErot = ½ ( miri2) 2
Moment of Inertia (I)
I = miri2
The kg . m2 is the SI unit for moment of inertia.
Moment of inertia is a measure of the resistance of an object to changes in
its rotational motion.
![Page 35: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/35.jpg)
Rotational Motion Linear Motion
I m
v
Substituting I,
KErot = ½ I2
KErot = ½ I2 KE = ½ mv2
![Page 36: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/36.jpg)
Example 1
Consider an oxygen molecule (O2) rotating in the x-y plane about the z-axis. The rotation axis passes through the center of the molecule,
perpendicular to its length. The mass of each oxygen atom is 2.66 x 10-26 kg, and at room
temperature the average separation between the two atoms is d = 1.21 x 10-10 m. (The atoms are
modeled as particles.)
a) Calculate the moment of inertia of the molecule about the z-axis.
![Page 37: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/37.jpg)
b) If the angular speed of the molecule about the z-axis is 4.60 x 1012 rad/s, what is its
rotational kinetic energy ?
![Page 38: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/38.jpg)
Lesson 5 : Calculation of Moments of Inertia
Moment of inertia of a rigid object is evaluated by dividing the object into many small volume elements, each
with mass = mi.
Since = m/V,
dm = dV
I = r2 dV
I = lim ri2mi = r2 dm
mi 0
![Page 39: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/39.jpg)
Find the moment of inertia of a uniform thin hoop of mass M and radius R about an axis perpendicular to the plane of the
hoop and passing through its center.
Example 1
![Page 40: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/40.jpg)
Calculate the moment of inertia of a uniform rigid rod of length L and mass M about an axis perpendicular to the
rod (the y-axis) and passing through its center of mass.
Example 2
![Page 41: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/41.jpg)
A uniform solid cylinder has a radius R, mass M, and length L. Calculate its
moment of inertia about its central axis (the z-axis).
Example 3
![Page 42: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/42.jpg)
Parallel-Axis Theorem
The moment of inertia about any axis parallel to and a distance D away from
this axis is
I = ICM + MD2
![Page 43: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/43.jpg)
Consider once again the uniform rigid rod of mass M and length L. Find the moment
of inertia of the rod about an axis perpendicular to the rod through one end
(the y’ axis).
Example 4
![Page 44: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/44.jpg)
Moment of Inertia of a Thin Cylindrical Shell (Hoop)
![Page 45: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/45.jpg)
Moment of Inertia of a Hollow Cylinder
![Page 46: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/46.jpg)
Moment of Inertia of a Solid Cylinder (Disk)
![Page 47: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/47.jpg)
Moment of Inertia of a Rectangular Plate
![Page 48: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/48.jpg)
Moment of Inertia of a Long Thin Rod
(Axis Through Center)
![Page 49: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/49.jpg)
Moment of Inertia of a Long Thin Rod
(Axis Through End)
![Page 50: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/50.jpg)
Moment of Inertia of a Solid Sphere
![Page 51: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/51.jpg)
Moment of Inertia of a Thin Spherical Shell
![Page 52: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/52.jpg)
The component Fsin tends to
rotate the wrench about
point O.
Lesson 6 : Torque
The tendency of a force to rotate an object about some axis is measured by a
vector quantity called torque.
![Page 53: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/53.jpg)
= r Fsin = Fd
distance between pivot point and point of application of F
perpendicular distance from pivot point to the
line of action of F
Torque should not be confused with force.
Torque has units of force x length or N.m. (Same as work but not called Joules.)
![Page 54: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/54.jpg)
F1 tends to rotate counterclockwise (+)
F2 tends to rotate clockwise (-)
= 1 + 2 = F1d1 – F2d2
The net torque about axis 0 is
![Page 55: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/55.jpg)
A one-piece cylinder is shaped as shown with a core section
protruding from the larger drum. The cylinder is free to rotate about
the central axis shown in the drawing. A rope wrapped around the drum, which has radius R1, exerts a force T1 to the right on the cylinder. A rope wrapped
around the core, which has radius R2, exerts a force T2 downward on
the cylinder.
Example 1
a) What is the net torque acting on the cylinder about the rotation axis (z-axis) ?
![Page 56: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/56.jpg)
b) Suppose T1 = 5.0 N, R1 = 1.0 m, T2 = 15.0 N, and R2 = 0.50 m. What is the net torque about the rotation axis, and which way does the cylinder rotate starting from rest ?
![Page 57: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/57.jpg)
Lesson 7 : Relationship between Torque and Angular Acceleration
Ft = mat
= Ftr = (mat)r
= (mr)r = (mr2)
Since I = mr2,
= I
(rotational analog to Newton’s Second Law)
![Page 58: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/58.jpg)
A uniform rod of length L and mass M is attached at one end to a frictionless pivot and is free to rotate about the pivot in the vertical
plane. The rod is released from rest in the horizontal position. What is the initial angular
acceleration of the rod and the initial linear acceleration of its right end ?
Example 1
![Page 59: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/59.jpg)
A wheel of radius R, mass M, and moment of inertia I
is mounted on a frictionless horizontal
axis. A light cord wrapped around the wheel supports
an object of mass m.
Example 2
a) Calculate the angular acceleration of the wheel.
![Page 60: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/60.jpg)
b) Calculate the linear acceleration of the object.
c) Calculate the tension in the cord.
![Page 61: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/61.jpg)
Work done by F on an object as it rotates
through an infinitesimal distance ds = r d
Lesson 8 : Work, Power, and Energy in Rotational Motion
dW = F.ds = (Fsin)r d
(The radial component of F does no work because it is perpendicular to the
displacement.)
![Page 62: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/62.jpg)
Since = rFsin,
Rate at which work is being done as an object rotates through angle d in
time interval dt is
dW
dt=
ddt
dW = d
(Power delivered to a rotating rigid object.)
P =dW
dt=
analogous to P = Fv in
linear motion
analogous to dW = Fx dx in linear motion
![Page 63: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/63.jpg)
= I= Iddt
= Idd
= Idd
ddt
(chain rule)
d = I d
Since dW = d,
dW = I d
![Page 64: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/64.jpg)
Integrating to find total work done,
W = i
f
I d = ½ If2 – ½ Ii
2
Work–KE Theorem for Rotational Motion
The net work done by external forces in rotating a symmetric rigid object about a
fixed axis equals the change in the object’s rotational energy.
![Page 65: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/65.jpg)
![Page 66: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/66.jpg)
A uniform rod of length L and mass M is free to rotate on a
frictionless pin passing through one
end. The rod is released from rest in
the horizontal position.
Example 1
a) What is its angular speed when it reaches its lowest position ?
![Page 67: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/67.jpg)
b) Determine the tangential speed of the center of mass and the tangential speed of the lowest point on the rod when it is in the vertical position.
![Page 68: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/68.jpg)
Consider two cylinders having different masses m1 and m2,
connected by a string passing over a pulley. The pulley has a
radius R and moment of inertia I about its axis of
rotation. The string does not slip on the pulley, and the
system is released from rest. Find the linear speeds of the
cylinders after cylinder 2 descends through a distance h, and the angular speed of
the pulley at this time.
Example 2
![Page 69: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/69.jpg)
A light string that is attached to a large block of mass 4m passes over a pulley with negligible rotational inertia and is wrapped around a vertical pole of radius r, as shown in
Experiment A above. The system is released from rest, and as the block descends the string unwinds and the vertical pole with its attached apparatus rotates. The
apparatus consists of a horizontal rod of length 2L, with a small block of mass m attached at each end. The
rotational inertia of the pole and the rod are negligible.
Example 3 : AP 2001 #3
![Page 70: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/70.jpg)
a) Determine the rotational inertia of the rod-and-block apparatus attached to the top of the
pole.
b) Determine the downward acceleration of the large block.
![Page 71: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/71.jpg)
c) When the large block has descended a distance D, how does the instantaneous total kinetic energy of the three blocks compare with the value 4mgD ? Check the appropriate space below.
____ Greater than 4mgD
____ Equal to 4mgD
____ Less than 4mgD
Justify your answer.
![Page 72: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/72.jpg)
The system is now reset. The string is rewound around the pole to bring the large block back to its
original location. The small blocks are detached from the rod and then suspended from each end of the
rod, using strings of length l. The system is again released from rest so that as the large block
descends and the apparatus rotates, the small blocks swing outward, as shown in Experiment B above. This time the downward acceleration of the block decreases with time after the system is released.
![Page 73: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/73.jpg)
d) When the large block has descended a distance D, how does the instantaneous total kinetic energy of the three blocks compare to that in part c) ? Check the appropriate space below.
____ Greater
____ Equal
____ Less
Justify your answer.
![Page 74: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/74.jpg)
Cylinder rolling on a straight path.
Lesson 9 : Rolling Motion of a Rigid Object
Center moves in a straight line (green line).
A point on the rim moves in a path called a cycloid (red curve).
![Page 75: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/75.jpg)
Speed of CM of Cylinder Rolling without Slipping
vCM =ds
dt
Since s = R,
vCM =ddt
R
vCM = R
for pure rolling motion only
![Page 76: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/76.jpg)
Acceleration of CM of Cylinder Rolling without Slipping
aCM =dvCM
dt
aCM =ddt
R
aCM = R
![Page 77: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/77.jpg)
Total Kinetic Energy of a Rolling Cylinder
KErot = ½ IP2
Since IP = ICM + MR2, (parallel-axis theorem)
KErot = ½ (ICM + MR2)2
OR
KErot = ½ ICM2 + ½ MR22
Since vCM = R,
KErot = ½ ICM2 + ½ MvCM2
![Page 78: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/78.jpg)
KEtotal = ½ ICM2 + ½ MvCM2
rotational KE translational KE
The total kinetic energy of a rolling object is the sum of the rotational kinetic energy about the center of mass and the translational kinetic
energy of the center of mass.
![Page 79: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/79.jpg)
![Page 80: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/80.jpg)
Three objects of uniform density – a solid sphere, a solid cylinder, and a hollow cylinder – are placed at the top of an incline. They are
all released from rest at the same elevation and roll without slipping. Which object reaches the
bottom first ? Which reaches it last ?
Example 1
![Page 81: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/81.jpg)
For the solid sphere shown above, calculate the linear speed of the center of mass at the bottom of the incline and the magnitude of the linear acceleration
of the center of mass.
Example 2
![Page 82: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/82.jpg)
An inclined plane makes an angle of with the horizontal, as shown above. A solid sphere of radius
R and mass M is initially at rest in the position shown, such that the lowest point of the sphere is a
vertical height h above the base of the plane. The sphere is released and rolls down the plane without slipping. The moment of inertia of the sphere about an axis through its center is 2/5 MR2. Express your
answers in terms of M, R, h, g, and .
Example 3 : AP 1986 #2
![Page 83: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/83.jpg)
a) Determine the following for the sphere when it is at the bottom of the plane.
i. Its translational kinetic energy
ii. Its rotational kinetic energy
![Page 84: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/84.jpg)
b) Determine the following for the sphere when it is on the plane.
i. Its linear acceleration
ii. The magnitude of the frictional force acting on it
![Page 85: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/85.jpg)
The solid sphere is replaced by a hollow sphere of identical radius R and mass M. The hollow sphere, which is released from the same location as the solid sphere, rolls down the incline without slipping.
c) What is the total kinetic energy of the hollow sphere at the bottom of the plane
![Page 86: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/86.jpg)
d) State whether the rotational kinetic energy of the hollow sphere is greater than, less than, or equal to that of the solid sphere at the bottom of the plane. Justify your
answer.
![Page 87: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/87.jpg)
Lesson 10 : Angular Momentum
F =
From linear motion :
dp
dt
Take cross product of each side with r :
r x F = r x dp
dt
Net Torque
= r x dp
dt
![Page 88: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/88.jpg)
x p is zero since = v, dr
dt
dr
dt
and v and p are parallel.
Add x p to right-hand side :dr
dt
= r x dp
dt+
dr
dtx p
=d(r x p)
dt
Analogous to dp in
translational motion angular momentum (L)
L = r x p
![Page 89: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/89.jpg)
Instantaneous Angular Momentum
The instantaneous angular momentum L of a particle relative to the origin O is defined by the cross
product of the particle’s instantaneous position vector r and
its instantaneous linear momentum p.
L = r x p
The SI unit of angular momentum is kg . m2/s.
![Page 90: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/90.jpg)
Since L = r x p,
=dL
dt rotational analog of Newton’s second law
The torque acting on a particle is equal to the time rate of change of the particle’s angular momentum.
![Page 91: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/91.jpg)
The direction of L is always perpendicular to the plane formed by r and p.
Since p = mv,
L = mvr sin
![Page 92: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/92.jpg)
A particle moves in the xy plane in a circular path of radius r, as shown above. Find the magnitude
and direction of its angular momentum relative to O when its linear velocity is v.
Example 1
![Page 93: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/93.jpg)
Angular Momentum of a System of Particles
The total angular momentum of a system of particles about some point is defined as the vector sum of the angular
momenta of the individual particles.
Ltot = L1 + L2 + …. + Ln = Li
Differentiating with respect to time :
dLtot
dt=
dLi
dt
![Page 94: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/94.jpg)
The net external torque acting on a system about some axis passing through an origin in an inertial frame equals the time rate of change of the total angular momentum of
the system about that origin.*
=
dLtot
dt
* This theorem applies even if the center of mass is accelerating, as long as and L are
evaluated relative to the center of mass.
![Page 95: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/95.jpg)
A sphere of mass m1 and a block of mass m2 are connected by a light cord that passes over a pulley, as
shown above. The radius of the pulley is R, and the mass of the rim is M. The spokes of the pulley have negligible mass. The block slides on a frictionless, horizontal surface. Find an expression for the linear
acceleration of the two objects, using the concepts of angular momentum and torque.
Example 2
![Page 96: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/96.jpg)
Each particle of this rigid object rotates in the xy plane about the z axis with angular speed .
Lesson 11 : Angular Momentum of a Rotating Rigid Object
Since v = r
L = mr2
Since I = mr2
L = I
![Page 97: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/97.jpg)
L = I
Differentiating with respect to time :(I is constant for a rigid object)
dL
dt= I
ddt
= I
=dL
dtSince
= Irotational form
of Newton’s second law
![Page 98: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/98.jpg)
Estimate the magnitude of the angular momentum of a bowling ball spinning at
10 rev/s.
Example 1
![Page 99: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/99.jpg)
A father of mass mf and daughter of mass md sit on opposite ends of a
seesaw at equal distances from the
pivot at the center. The seesaw is modeled as a rigid rod of mass M and length l and is pivoted
without friction. At a given moment, the
combination rotates in a vertical plane with an
angular speed .
Example 2
![Page 100: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/100.jpg)
a) Find an expression for the magnitude of the system’s angular momentum.
![Page 101: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/101.jpg)
b) Find an expression for the magnitude of the angular acceleration of the system when the seesaw makes an angle with the horizontal.
![Page 102: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/102.jpg)
A uniform solid cylinder of mass m1 and radius R is mounted on frictionless bearings about a fixed axis
through O. The moment of inertia of the cylinder about the axis is I = ½ m1R2. A block of mass m2,
suspended by a cord wrapped around the cylinder as shown above, is released at time t = 0.
Example 3 : AP 1983 #2
![Page 103: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/103.jpg)
a) On the diagram below draw and identify all of the forces acting on the cylinder and on the block.
![Page 104: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/104.jpg)
b) In terms of m1, m2, R, and g, determine each of the following.
i) The acceleration of the block.
![Page 105: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/105.jpg)
iii) The angular momentum of the disk as a function of t.
ii) The tension in the cord.
![Page 106: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/106.jpg)
A system consists of two small disks, of masses m and 2m, attached to a rod of negligible mass of length 3l as shown above. The rod is free to turn about a vertical axis through
point P. The two disks rest on a rough horizontal surface; the coefficient of friction between the disks and the surface is . At time t = 0, the rod has an initial counterclockwise angular velocity o about P. The system is gradually brought to rest by friction. Develop expressions for the following quantities
in terms of , m, l, g, and o.
Example 4 : AP 1982 #3
![Page 107: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/107.jpg)
a) The initial angular momentum of the system about the axis through P.
b) The frictional torque acting on the system about the axis through P.
![Page 108: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/108.jpg)
c) The time T at which the system will come to rest.
![Page 109: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/109.jpg)
Example 5 : AP 1996 #3
Consider a thin uniform rod of mass M and length l , as shown above.
l
M
a) Show that the rotational inertia of the rod about an axis through its center and
perpendicular to its length is Ml 2/12.
![Page 110: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/110.jpg)
The rod is now glued to a thin hoop of mass M and radius R = l /2 to form a rigid assembly, as shown above. The centers of the rod and hoop coincide at point P. The assembly is mounted
on a horizontal axle through point P and perpendicular to the page.
![Page 111: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/111.jpg)
b) What is the rotational inertia of the rod-hoop assembly about the axle ?
![Page 112: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/112.jpg)
Several turns of string are wrapped tightly around the circumference of the hoop. The system is at rest when a cat, also of mass M, grabs the free end of the string and hangs vertically from it
without swinging as it unwinds, causing the rod-hoop assembly to rotate. Neglect friction and the
mass of the string.
c) Determine the tension T in the string.
![Page 113: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/113.jpg)
d) Determine the angular acceleration of the rod-hoop assembly.
e) Determine the linear acceleration of the cat.
![Page 114: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/114.jpg)
f) After descending a distance H = 5 l /3, the cat lets go of the string. At that instant, what is the
angular momentum of the cat about point P ?
![Page 115: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/115.jpg)
Lesson 12 : Conservation of Angular Momentum
The total angular momentum of a system is constant in both magnitude and
direction if the resultant external torque acting on the system is zero, that is, if the
system is isolated.
=
dLtot
dt= 0,Since Ltot = constant
Lbefore = Lafter
![Page 116: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/116.jpg)
When arms are moved inward, I
decreases.
Since I remains constant, as I
decreases, must increase.
Lbefore = Lafter
Since L = I
Iii = Iff = constant
![Page 117: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/117.jpg)
A horizontal platform in the shape of a circular disk rotates freely in a horizontal plane about a frictionless
vertical axle. The platform has a mass M = 100 kg and a radius R = 2.0 m. A student whose mass is m = 60 kg
walks slowly from the rim of the disk toward its center. If the angular speed of the system is 2.0 rad/s when the student is at the rim, what is the angular speed when he
reaches a point r = 0.50 m from the center ?
Example 1
![Page 118: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/118.jpg)
In a favorite classroom demonstration, a student holds the
axle of a spinning bicycle wheel while seated on a stool that is free to rotate. The student and stool are
initially at rest while the wheel is spinning in a horizontal plane with an initial angular momentum Li that points upward. When the wheel is inverted about its center by 180o,
the student and stool start rotating. In terms of Li, what are the
magnitude and the direction of L for the student plus stool ?
Example 2
![Page 119: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/119.jpg)
A 2.0 kg disk traveling at 3.0 m/s strikes a 1.0 kg stick of length 4.0 m that is lying flat on nearly frictionless ice. Assume that the collision is elastic and that the disk does not deviate from its original line of motion. Find the translational speed of the disk, the translational
speed of the stick, and the angular speed of the stick after the collision. The moment of inertia of the stick
about its center of mass is 1.33 kg.m2.
Example 3
![Page 120: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/120.jpg)
A 1.0 kg object is moving horizontally with a velocity of 10 m/s, as shown above, when it makes a glancing
collision with the lower end of a bar that was hanging vertically at rest before the collision. For the system
consisting of the object and bar, linear momentum is not conserved in this collision, but kinetic energy is
conserved. The bar, which has a length l = 1.2 m and a mass m = 3.0 kg, is pivoted about the upper end.
Example 4 : AP 1987 #3
![Page 121: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/121.jpg)
Immediately after the collision the object moves with speed v at an angle relative to its original direction. The bar swings freely, and after the collision reaches a maximum angle of 90o with respect to the vertical. The moment of inertia of the bar about the pivot is
Ibar = ml 2/3. Ignore all friction.
a) Determine the angular velocity of the bar immediately after the collision.
![Page 122: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/122.jpg)
b) Determine the speed v of the 1 kg object immediately after the collision.
c) Determine the magnitude of the angular momentum of the object about the pivot just before the collision.
![Page 123: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/123.jpg)
d) Determine the angle .
![Page 124: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/124.jpg)
Two identical spheres, each of mass M and negligible radius, are fastened to opposite ends of a rod of negligible mass and length 2l. This system is initially at rest with the rod horizontal, as shown above, and is free to rotate about a frictionless, horizontal axis through the center of the rod and perpendicular to the plane of the page. A bug, of mass 3M, lands gently on the sphere on the left. Assume that the size of the bug is small compared to the length of the rod. Express your answers to all parts of the question in terms
of M, l , and physical constants.
Example 5 : AP 1992 #2
![Page 125: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/125.jpg)
a) Determine the torque about the axis immediately after the bug lands on the sphere.
b) Determine the angular acceleration of the rod-spheres-bug system immediately after the bug lands.
![Page 126: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/126.jpg)
The rod-spheres-bug system swings about the axis. At the instant that the rod is vertical, as
shown above, determine each of the following.
c) The angular speed of the bug.
![Page 127: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/127.jpg)
d) The angular momentum of the system.
e) The magnitude and direction of the force that must be exerted on the bug by the sphere to keep the bug from being thrown off the sphere.
![Page 128: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/128.jpg)
Lesson 13 : Rotational Equilibrium
A system is in rotational equilibrium if the net torque on it is zero about any axis.
= 0
Since = I
= 0
= 0 does not mean an absence of rotational motion. Object can be rotating at a constant
angular speed.
![Page 129: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/129.jpg)
Consider the object subject to the two
forces shown to the right. Choose the
correct statement with regard to this situation.
The object is in
Example 1
____ force equilibrium but not torque equilibrium.
____ torque equilibrium but not force equilibrium.
____ both force and torque equilibrium.
____ neither force nor torque equilibrium.
![Page 130: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/130.jpg)
Consider the object subject to the three forces shown to the
right. Choose the correct statement with regard to this situation.
The object is in
Example 2
____ force equilibrium but not torque equilibrium.
____ torque equilibrium but not force equilibrium.
____ both force and torque equilibrium.
____ neither force nor torque equilibrium.
![Page 131: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/131.jpg)
Center of Gravity
To compute the torque due to the gravitational force on an object of mass M, we need only consider the force Mg acting
at the center of gravity of the object.
Center of gravity = center of mass if g is constant over the object.
![Page 132: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/132.jpg)
A seesaw consisting of a uniform board of mass M and length l supports a father and daughter with masses mf and md, respectively. The support (called the fulcrum) is under the center of gravity of the board, the father is a
distance d from the center, and the daughter is a distance l /2 from the center.
Example 3
![Page 133: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/133.jpg)
a) Determine the magnitude of the upward force n exerted by the support on the board.
b) Determine where the father should sit to balance the system.
![Page 134: Rotational Motion and Angular Momentum Unit 6. A particle at point P at a fixed distance r from origin is rotating about axis O. Lesson 1 : Angular Position,](https://reader038.fdocuments.us/reader038/viewer/2022102906/56649cb05503460f94975527/html5/thumbnails/134.jpg)
A person holds a 50.0 N sphere in his hand. The forearm is horizontal, as shown above. The biceps muscle is attached 3.00 cm from the joint, and the sphere is 35.0 cm from the joint. Find the upward force exerted by the biceps on the
forearm and the downward force exerted by the upper arm on the forearm and acting at the joint. Neglect the weight
of the forearm.
Example 4