1. Lecture 18 Rotational Motion
General Physics I, Lec 18, By/ T. A. Eleyan

2. General Physics I, Lec 18, By/ T. A. Eleyan
Angular Position, θ
For circular motion, the distance (arc length) s, the radius r, and the angle  are related by:
q > 0 for counterclockwise rotation from reference line
Note that  is measured in radians:
General Physics I, Lec 18, By/ T. A. Eleyan

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The angular displacement is defined as the angle the object rotates through during some time interval
Every point on the disc undergoes the same angular displacement in any given time interval
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4. General Physics I, Lec 18, By/ T. A. Eleyan
Angular Velocity, ω
Notice that as the disk rotates,  changes. We define the angular displacement, , as:
 = f - i
which leads to the average angular speed wav
General Physics I, Lec 18, By/ T. A. Eleyan

5. Instantaneous Angular Velocity
As usual, we can define the instantaneous angular velocity as:
Note that the SI units of  are: rad/s = s-1
w > 0 for counterclockwise rotation
< 0 for clockwise rotation
If v = speed of a an object traveling around a circle of radius r
w = v / r
General Physics I, Lec 18, By/ T. A. Eleyan

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The period of rotation is the time it takes to complete one revolution.
Problem:
What is the period of the Earth’s rotation about its own axis?
What is the angular velocity of the Earth’s rotation about its own axis?
General Physics I, Lec 18, By/ T. A. Eleyan

7. General Physics I, Lec 18, By/ T. A. Eleyan
Angular Acceleration, a
We can also define the average angular acceleration aav:
And instantaneous angular acceleration
The SI units of a are: rad/s2 = s-2
We will skip any detailed discussion of angular acceleration, except to note that angular acceleration is the time rate of change of angular velocity
General Physics I, Lec 18, By/ T. A. Eleyan

8. Notes about angular kinematics:
When a rigid object rotates about a fixed axis, every portion of the object has the same angular speed and the same angular acceleration
i.e. q,w, and a are not dependent upon r, distance form hub or axis of rotation
General Physics I, Lec 18, By/ T. A. Eleyan

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Examples:
1. Bicycle wheel turns 240 revolutions/min. What is its angular velocity in radians/second?
2. If wheel slows down uniformly to rest in 5 seconds, what is the angular acceleration?
General Physics I, Lec 18, By/ T. A. Eleyan

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3. How many revolution does it turn in those 5 sec?
Recall that for linear motion we had:
Perhaps something similar for angular quantities?
General Physics I, Lec 18, By/ T. A. Eleyan

11. Analogies Between Linear and Rotational Motion
Rotational Motion About a Fixed Axis with Constant Acceleration
Linear Motion with Constant Acceleration
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12. Relationship Between Angular and Linear Quantities
Displacements
Speeds
Accelerations
General Physics I, Lec 18, By/ T. A. Eleyan

13. General Physics I, Lec 18, By/ T. A. Eleyan
Tangential and the radial acceleration.
Since the tangential speed v is
The magnitude of tangential acceleration at is
The radial or centripetal acceleration ar is
Total linear acceleration is
General Physics I, Lec 18, By/ T. A. Eleyan

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Example: (a) What is the linear speed of a child seated 1.2m from the center of a steadily rotating merry-go-around that makes one complete revolution in 4.0s? (b) What is her total linear acceleration?
Since the angular speed is constant, there is no angular acceleration.
Tangential acceleration is
Radial acceleration is
General Physics I, Lec 18, By/ T. A. Eleyan

15. Calculation of Moments of Inertia
Moments of inertia for large objects can be computed, if we assume that the object consists of small volume elements with mass, Dmi.
The moment of inertia for the large rigid object is:
It is sometimes easier to compute moments of inertia in terms of volume of the elements rather than their mass
Using the volume density, r, replace dm in the above equation with dV.
The moments of inertia becomes
General Physics I, Lec 18, By/ T. A. Eleyan

16. General Physics I, Lec 18, By/ T. A. Eleyan
The moment of inertia of a uniform hoop of mass M and radius R about an axis perpendicular to the plane of the hoop and passing through its center.
The moment of inertia is
The moment of inertia for this object is the same as that of a point of mass M at the distance R. x y R O dm
General Physics I, Lec 18, By/ T. A. Eleyan

17. Example for Rigid Body Moment of Inertia
Calculate the moment of inertia of a uniform rigid rod of length L and mass M about an axis perpendicular to the rod and passing through its center of mass.
The line density of the rod is
so the mass is
General Physics I, Lec 18, By/ T. A. Eleyan

18. General Physics I, Lec 18, By/ T. A. Eleyan
Torque: The ability of a force to rotate a body about some axis.
Only the component of the force that is perpendicular to the radius causes a torque.
= r (Fsinq)
Equivalently, only the perpendicular distance between the line of force and the axis of rotation, known as the moment arm r, can be used to calculate the torque.
t = rF = (rsinq)F
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The net torque about a point O is the sum of all torques about O:
General Physics I, Lec 18, By/ T. A. Eleyan

20. General Physics I, Lec 18, By/ T. A. Eleyan
Problem:
Calculate the net torque on the 0.6-m rod about the nail at the left. Three forces are acting on the rod as shown in the diagram.
30°
0.3 m
4 N
5 N
6 N
General Physics I, Lec 18, By/ T. A. Eleyan

21. Torque & Angular Acceleration
Let’s consider a point object with mass m rotating on a circle.
The tangential force Ft is
The torque due to tangential force Ft is
Torque acting on a particle is proportional to the angular acceleration.
Analogs to Newton’s 2nd law of motion in rotation.
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22. General Physics I, Lec 18, By/ T. A. Eleyan
How about a rigid object?
The external tangential force dFt is
The torque due to tangential force Ft is
The total torque is
General Physics I, Lec 18, By/ T. A. Eleyan

23. General Physics I, Lec 18, By/ T. A. Eleyan
Example :
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 are the initial angular acceleration of the rod and the initial linear acceleration of its right end?
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24. General Physics I, Lec 18, By/ T. A. Eleyan
The only force generating torque is the gravitational force Mg
Since the moment of inertia of the rod when it rotates about one end
We obtain
Using the relationship between tangential and angular acceleration
The tip of the rod falls faster than an object undergoing a free fall.
General Physics I, Lec 18, By/ T. A. Eleyan

25. Rotational Kinetic Energy
Kinetic energy of a rigid object that undergoing a circular motion:
Kinetic energy of a mass mi, moving at a tangential speed, vi, is
Since a rigid body is a collection of mass, the total kinetic energy of the rigid object is
The moment of Inertia, I, is defined as
The above expression is simplified to
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26. General Physics I, Lec 18, By/ T. A. Eleyan
Example:
In a system consists of four small spheres as shown in the figure, assuming the radii are negligible and the rods connecting the particles are massless, compute the moment of inertia and the rotational kinetic energy when the system rotates about the y-axis at w. x y M l m b O
General Physics I, Lec 18, By/ T. A. Eleyan

27. General Physics I, Lec 18, By/ T. A. Eleyan
Since the rotation is about y axis, the moment of inertia about y axis, Iy, is
Why are some 0s?
This is because the rotation is done about y axis, and the radii of the spheres are negligible.
Thus, the rotational kinetic energy is
Find the moment of inertia and rotational kinetic energy when the system rotates on the x-y plane about the z-axis that goes through the origin O.
General Physics I, Lec 18, By/ T. A. Eleyan