Physics CHAPTER 8 ROTATIONAL MOTION. The Radian The radian is a unit of angular measure The radian...
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Transcript of Physics CHAPTER 8 ROTATIONAL MOTION. The Radian The radian is a unit of angular measure The radian...
Physics CHAPTER 8
ROTATIONAL MOTION
The Radian The radian is a unit
of angular measure The radian can be
defined as the arc length s along a circle divided by the radius r
sr
57.3
More About Radians
Comparing degrees and radians
Converting from degrees to radians
3.572
360rad1
]rees[deg180
]rad[
Angular Displacement
Axis of rotation is the center of the disk
Need a fixed reference line
During time t, the reference line moves through angle θ
Angular Displacement, cont.
The angular displacement is defined as the angle the object rotates through during some time interval
The unit of angular displacement is
the radian Each point on the object undergoes
the same angular displacement
fi
Average Angular Speed
The average angular speed, ω, of a rotating rigid object is the ratio of the angular displacement to the time interval
fiav
fit t t
Angular Speed, cont.
The instantaneous angular speed is defined as the limit of the average speed as the time interval approaches zero
Units of angular speed are radians/sec rad/s
Speed will be positive if θ is increasing (counterclockwise)
Speed will be negative if θ is decreasing (clockwise)
Average Angular Acceleration
The average angular acceleration, , of an object is defined as the ratio of the change in the angular speed to the time it takes for the object to undergo the change:
fiav
fit t t
Angular Acceleration, cont
Units of angular acceleration are rad/s² Positive angular accelerations are in the
counterclockwise direction and negative accelerations are in the clockwise direction
When a rigid object rotates about a fixed axis, every portion of the object has the same angular speed and the same angular acceleration
Angular Acceleration, final
The sign of the acceleration does not have to be the same as the sign of the angular speed
The instantaneous angular acceleration is defined as the limit of the average acceleration as the time interval approaches zero
Angular AccelerationAngular acceleration α measures how rapidly the angular velocity is changing:
Slide 7-17
Linear and Circular Motion Compared
Slide 7-18
Linear and Circular Kinematics Compared
Slide 7-19
Sign of the Angular Acceleration
Slide 7-20
Relationship Between Angular and Linear Quantities Displacements
Speeds
Accelerations
Every point on the rotating object has the same angular motion
Every point on the rotating object does not have the same linear motion
rs
tv r
ta r
Centripetal Acceleration and Angular Velocity
The angular velocity and the linear velocity are related (v = ωr)
The centripetal acceleration can also be related to the angular velocity
ra 2C
Vector Nature of Angular Quantities
Angular displacement, velocity and acceleration are all vector quantities
Direction can be more completely defined by using the right hand rule Grasp the axis of rotation
with your right hand Wrap your fingers in the
direction of rotation Your thumb points in the
direction of ω
Velocity Directions, Example
In a, the disk rotates clockwise, the velocity is into the page
In b, the disk rotates counterclockwise, the velocity is out of the page
Righty-tighty
Lefty-loosey
Centripetal and Tangential Acceleration
Slide 7-22
Force vs. Torque
Forces cause accelerationsTorques cause angular
accelerationsForce and torque are related
Torque
The door is free to rotate about an axis through O There are three factors that determine the
effectiveness of the force in opening the door: The magnitude of the force The position of the application of the force The angle at which the force is applied
Torque, cont Torque, , is the tendency of a
force to rotate an object about some axis= Fr
is the torqueF is the force
symbol is the Greek taur is the length of the position
vector SI unit is N.m
Interpreting Torque
rF rF sin
Torque is due to the component of the force perpendicular to the radial line.
Slide 7-25
Torque is a vector quantityThe direction is perpendicular to the plane determined by the position vector and the force
A Second Interpretation of Torque
rF rF sin
Slide 7-26
FsinFsin
Signs and Strengths of the Torque
Slide 7-27
If the turning tendency of the force is counterclockwise, the torque will be positive
If the turning tendency is clockwise, the torque will be negative
Multiple Torques
When two or more torques are acting on an object, the torques are addedAs vectors
If the net torque is zero, the object’s rate of rotation doesn’t change
General Definition of Torque
The applied force is not always perpendicular to the position vector
The component of the force perpendicular to the object will cause it to rotate
When the force is parallel to the position vector, no rotation occurs
When the force is at some angle, the perpendicular component causes the rotation
General Definition of Torque, final
Taking the angle into account leads to a more general definition of torque: Fr sin
F is the forcer is the position vector is the angle between the force
and the position vector
Lever Arm
The lever arm, d, is the perpendicular distance from the axis of rotation to a line drawn along the direction of the force
d = r sin
Net Torque
The net torque is the sum of all the torques produced by all the forces Remember to account for the
direction of the tendency for rotationCounterclockwise torques are
positiveClockwise torques are negative
Checking UnderstandingThe four forces shown have the same strength. Which force would be most effective in opening the door?
A. Force F1
B. Force F2
C. Force F3
D. Force F4
E. Either F1 or F3Slide 7-23
AnswerThe four forces shown have the same strength. Which force would be most effective in opening the door?
A. Force F1
B. Force F2
C. Force F3
D. Force F4
E. Either F1 or F3Slide 7-24
Moment of Inertia
The moment of inertia, I, of a point mass is equal to mass of the object times the square of the distance from the object’s axis of rotation.
SI units are kg m2
Applying Newton’s 2nd Law results in
2I m r
Moment of Inertia
The moment of inertia of an object is the rotational equivalent to the mass of the object in a linear motion.
Ex. For linear motion, the heavier the mass the more difficult it is to get it to move. In rotational motion, a high I, moment of inertia, means that it is difficult to get the object to rotate on an axis.
The size of the moment of inertia, depends on the radius of rotation from the center axis and the distribution of mass around the axis of rotation.
Moment of Inertia If the radius length is large, it
will be more difficult to get the mass to rotate which indicates a higher moment of inertia. So if we apply a torque closer to the axis of rotation, it will be easier to cause the rotation to occur.
If the mass is distributed closer to the axis of rotation, the rotation will be easier to start and the Moment of Inertia will be less
I1 > I2
Moments of Inertia for Various ObjectsObject Location of Axis Diagram Moment of Inertia
Equation
Thin hoop of radius r
Through central diameter
Solid uniform cylinder of radius r
Through center
Uniform Sphere of radius r
Through center
Long uniform rod of length l
Through center
Long uniform rod of length l
Through end
Thin rectangular plane of length l and width w
Through center
Newton’s Second Law for a Rotating Object
The angular acceleration is directly proportional to the net torque
The angular acceleration is inversely proportional to the moment of inertia of the object
I
More About Moment of Inertia
There is a major difference between moment of inertia and mass: the moment of inertia depends on the quantity of matter and its distribution in the rigid object.
The moment of inertia also depends upon the location of the axis of rotation
Moment of Inertia of a Uniform Ring Image the hoop is
divided into a number of small segments, m1 …
These segments are equidistant from the axis
2 2i iI m r M R
Other Moments of Inertia
Example, Newton’s Second Law for Rotation Draw free body
diagrams of each object Only the cylinder is
rotating, so apply = I
The bucket is falling, but not rotating, so apply F = m a
Remember that a = r and solve the resulting equations
Torque and Equilibrium
First Condition of Equilibrium The net external force must be zero
This is a necessary, but not sufficient, condition to ensure that an object is in complete mechanical equilibrium
This is a statement of translational equilibrium
0
0 0x y
or
and
F
F F
Torque and Equilibrium, cont
To ensure mechanical equilibrium, you need to ensure rotational equilibrium as well as translational
The Second Condition of Equilibrium states The net external torque must be zero
0
Equilibrium Example
The woman, mass m, sits on the left end of the see-saw
The man, mass M, sits where the see-saw will be balanced
Apply the Second Condition of Equilibrium and solve for the unknown distance, x
Axis of Rotation
If the object is in equilibrium, it does not matter where you put the axis of rotation for calculating the net torque The location of the axis of rotation is
completely arbitrary Often the nature of the problem will
suggest a convenient location for the axis When solving a problem, you must specify
an axis of rotation Once you have chosen an axis, you must
maintain that choice consistently throughout the problem
Notes About Equilibrium
A zero net torque does not mean the absence of rotational motion An object that rotates at uniform
angular velocity can be under the influence of a zero net torqueThis is analogous to the
translational situation where a zero net force does not mean the object is not in motion
Solving Equilibrium Problems
Draw a diagram of the system Include coordinates and choose a
rotation axis
Isolate the object being analyzed and draw a free body diagram showing all the external forces acting on the object For systems containing more than one
object, draw a separate free body diagram for each object
Problem Solving, cont.
Apply the Second Condition of Equilibrium This will yield a single equation, often with one
unknown which can be solved immediately
Apply the First Condition of Equilibrium This will give you two more equations
Solve the resulting simultaneous equations for all of the unknowns Solving by substitution is generally easiest
Example of a Free Body Diagram (Forearm)
Isolate the object to be analyzed Draw the free body diagram for that object
Include all the external forces acting on the object
Example of a Free Body Diagram (Beam) The free body
diagram includes the directions of the forces
The weights act through the centers of gravity of their objects
Fig 8.12, p.228
Slide 17
Example of a Free Body Diagram (Ladder)
The free body diagram shows the normal force and the force of static friction acting on the ladder at the ground
The last diagram shows the lever arms for the forces
Center of Gravity
The force of gravity acting on an object must be considered
In finding the torque produced by the force of gravity, all of the weight of the object can be considered to be concentrated at a single point
Calculating the Center of Gravity The object is
divided up into a large number of very small particles of weight (mg)
Each particle will have a set of coordinates indicating its location (x,y)
Calculating the Center of Gravity, cont.
We assume the object is free to rotate about its center
The torque produced by each particle about the axis of rotation is equal to its weight times its lever armFor example, m1 g x1
Calculating the Center of Gravity, cont.
We wish to locate the point of application of the single force whose magnitude is equal to the weight of the object, and whose effect on the rotation is the same as all the individual particles.
This point is called the center of gravity of the object
Coordinates of the Center of Gravity
The coordinates of the center of gravity can be found from the sum of the torques acting on the individual particles being set equal to the torque produced by the weight of the object
i i i icg cg
i i
m x m yx and y
m m
Center of Gravity of a Uniform Object
The center of gravity of a homogenous, symmetric body must lie on the axis of symmetry.
Often, the center of gravity of such an object is the geometric center of the object.