Department of Physics and Applied Physics95.141, Fall 2013, Lecture 17
Course website:http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsI
Lecture Capture: http://echo360.uml.edu/danylov2013/physics1fall.html
Lecture 17
Chapter 10
Rotational MotionTorque
11.13.2013Physics I
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 17
Chapter 10
Rotational Motion Rotational kinematics Rotational dynamics Torque
Outline
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 17
Exam2 Results
295 students took the examAverage 51.8/100
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 17
Exam2 Results
295 students took the examAverage 51.8/100
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 17
Rotational MotionIn addition to translation, extended objects can rotate
Need to develop a vocabulary for
describing rotational motion
There is rotation everywhere you look in the universe, from the nuclei of atoms to spiral galaxies
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 17
Polar coordinates for circular motion
Rectangular coordinates are well suited to describing motion in a straight line. However, rectangular coordinates are not useful for describing circular motion.
Since circular motion plays a prominent role in physics, it is worth introducing a special coordinate system –POLAR coordinate system.
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 17
Angular Position in polar coordinatesConsider a pure rotational motion:
an object moves around a fixed axis.
x
arclength
R
R, θ in radians!
y
R
We define object’s position with: R,
R
So, one radian is defined as the angle subtendedby an arc whose length is equal to the radius.
Rrad
1
The radian is dimensionless, so there is no need to mention it in calculations
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 17
Angular Position in polar coordinates
x
R
rad2
radians!
y
R
length arc
ApplyR
R 2/
2
x
R R
RR2
rad2360
3.572/3601 rad
2/4R2 R
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 17
Use Radians to get an arclength
R
R 60
R3
R
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 17
The average angular velocity is defined as the total angular displacement divided by time:
Angular displacement
12 Angular displacement:
The instantaneous angular velocity:
t
dtd
tt
0
lim
is the same for all points of a rotating object.
The angular velocity of any
point on a solid object rotating about a
fixed axis is the same. Both
Bonnie and Klyde go around one
revolution (2 radians) every 2 seconds.
ConcepTest 1 Bonnie and Klyde
BonnieKlyde
Bonnie sits on the outer rim of a merry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one complete revolution every2 seconds.Klyde’s angular velocity is:
A) same as Bonnie’sB) twice Bonnie’sC) half of Bonnie’sD) one-quarter of Bonnie’sE) four times Bonnie’s
t
sec21rev
secsec2
2 radrad is the same for both
rabbits
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 17
12
12
ttt
Angular Acceleration
Average angular acceleration:
Instantaneous angular acceleration: dtd
tt
0
lim
The angular acceleration is the rate at which the angular velocity changes with time:
121t2t
Since is the same for all points of a rotating object, angular acceleration also will be the same for all points.
Thus, and α are properties of a rotating object
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 17
Vector Nature of Angular Quantities
However, direction of rotation can be clockwise and counterclockwise. How can we show a difference between them?
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 17
Vector of Angular VelocityBoth ω and α can be treated as vectors:
(by convention)Right Hand Rule
Curl fingers on right hand to trace rotation of object
Direction of thumb is vector direction for angular velocity.
+z
x
y
)ˆ( k
)ˆ( k+z
x
y
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 17
Vector of Angular acceleration
the object will be "speeding up" if the angular acceleration is in the same direction as the angular velocity, and
the object will be "slowing down" if the angular acceleration is in the opposite direction of the angular velocity.
Just as was the case for linear motion:
i
f
f
i
0 0
0
if
if
tt
0
if
if
tt
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 17
Relation between linear and angular velocities
Each point on a rotating rigid body has the same angular displacement, velocity, and acceleration!
The corresponding linear (or tangential) variables depend on the radius and the linear velocity is greater for points farther from the axis.
vtan ddt
R
d
dltanv
RIn the 1st slide, we defined:
So we can write: Rdd vtan R d
dt R
Rv tan
Relation between linear and angular velocities ( in rad/sec)
By definition, linear velocity:
Remember it!!!!You will use it often!!!
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 17
Example: Linear/angular velocityRope wound around a circular cylinder unwraps without stretching or slipping, its speed and acceleration at any instant are equal to the speed and tangential acceleration of the point at which it is tangent to the cylinder
tanv
v
Rvv tan
But their linear speeds v will be
different because and
Bonnie is located farther out (larger
radius R) than Klyde.
Bonnie
Klyde
BonnieKlyde V21V
ConcepTest 2 Bonnie and Klyde IIA) KlydeB) BonnieC) both the sameD) linear velocity is zero
for both of them
Bonnie sits on the outer rim of a merry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one revolution every 2 seconds. Who has the larger linear (tangential) velocity?
Rv tan
We already know that all points of a rotating body have the same angular velocity .
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 17
Relation between linear and angular acceleration
dtdva tan
tan
dtdRa
tan R
atotal atan
aR
RvaR
2tan
aR
tana
Rv tanBy definition of linear acceleration: Let’s use:
Ra tan
Total acceleration
Finally, recall that any object that is undergoing circular motion experiences an inwardly directed radial acceleration (centripetal one).
Rv tanR2
totala
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 17
Rotational kinematic equationsThe equations of motion for translational and rotational motion
(for constant acceleration) are identical
o ot 12t2
)(221
22
2
oo
oo
o
xxavv
attvxx
atvv
oo 222
o tv
a
x
Translational kinematic equations
Rotational kinematic equations
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 17
Rotational Dynamics
What causes rotation?
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 17
Torque is a turning force (the rotational equivalent of force).
It depends on force, lever arm, angle:
Why? What causes rotation?
TorqueWhen we apply the force, the door turns on its hinges(a turning effect is produced). In the 1st case, we are able to open the door with ease. In the 2nd case, we have to apply much more force to cause the same turning effect.
A longer lever arm is very helpful in rotating objects.
rF sinF
r
Torque due to a force F applied at a distance r from the pivot, at an angle θ to the radial line.
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 17
Axis of rotation
There are two ways of calculating Torque
rF sin
r(F sin )
F(rsin )
F
rsinF
Let’s arrange it like this:
Perpendicular component of force acting at a distance r from the axis
Axis
F
r
sinrOr Let’s arrange it like this:
Force times arm lever extending from the axis to the line of force and perpendicular to the line of force
1
2
You are using a wrench to loosen a rusty nut. Which arrangement will be the most effective in loosening the nut?
A
C D
B
Because the forces are all the same, the only difference is the lever arm. The arrangement with the largest lever arm (case #2) will provide the largest torque.
E) all are equally effective
Follow-up: What is the difference between arrangement A and D?
ConcepTest 3 Using a Wrench
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 17
Torque causes angular acceleration:
Force causes linear acceleration: (N.2nd law):
Newton’s 2nd law of rotation
I
I is the Moment of Inertia (rotational equivalent of mass)
amF
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 17
Thank youSee you on Monday
ConcepTest 3 Closing a DoorA
B
C
D
E
The diagram shows the top view of a door, hinge to the left and door-knob to the right. The same force F is applied differently to the door. In which case is the turning ability provided by the applied force about the rotation axis greatest?
The torque is t = Fd sinq, and so the force that is at 90° to the lever arm is the one that will have the largest torque. (Clearly, to close the door, you want to push perpendicularly!!) So A or B? B has larger lever arm
A B C
D
E
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 17
Frequency and Period
We can relate the angular velocity of rotation to the frequency of rotation:
1 rev/s =2 rad/s
f 1T
2
1 hertz (Hz) = 1 rev/s
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