1. HW #8 due Thursday, Nov 4 at 11:59 p.m.
Recitation Quiz #8 tomorrow
Last Time: Center of Gravity, Equilibrium Problems
Today: Moment of Inertia, Torque and Angular Acceleration
B. Plaster is in contact this week. Send with any questions and/or concerns. His office hours are cancelled this week.

2. Recall: The See-Saw m
Q: Suppose our familiar see-saw is initially perfectly balanced (not rotating). You place a mass m on one end. What happens ?
A: We know that the see-saw rotates about the pivot, due to the torque of the weight mg about the center !
Summary:
The see-saw initially had an angular speed of zero. The torque caused the see-saw to acquire some non-zero angular speed. A change in the angular speed means there was an angular acceleration.  The torque caused an angular acceleration.

3. Torque and Angular Acceleration
An object of mass m connected to a massless rod of length r is rotating horizontally.
A tangential force Ft acts perpendicular to the rod, causing a tangential acceleration (i.e., an increase in the angular speed and tangential velocity).
By Newton’s Second Law: 1 4
But Ft∙r is the torque !
Multiply by r : 2
α is proportional to τ 3
Recall: at = rα
Proportionality constant mr2 called “moment of inertia”

4. Torque and Angular Acceleration
Consider a RIGID rotating disc.
Think of the disc as made up of a LARGE number of particles, with mass m1, m2, m3, etc.
Suppose the disc is undergoing an angular acceleration α.
Disc is rigid  All of these masses are undergoing the same acceleration α. The net torque on the disc is:
Moment of Inertia
Define:
[ Analog to ΣF = ma !! ]

5. Rotational vs. Linear Dynamics
For a given net force:
For a given net torque:
The larger the mass m, the smaller the magnitude of a (“inertia”).
The larger the moment of inertia I, the smaller the magnitude of α (“rotational inertia”).

6. Moments of Inertia
Moment of inertia of a single object rotating about some axis is: m
r : distance from axis of rotation r
[ SI: kg-m2 ]
For a composite object (made up of many masses):
Key difference between ΣF = ma and Στ = Iα :
In ΣF = ma, m does not depend on how the mass is distributed. m DOES NOT change depending on its location.
In Στ = Iα, I DOES depend on how the mass is distributed I depends on the distance of each mass from the rotation axis.

7. Example
Consider a baton to be made up of a 1.0-m long massless rod, with two 1.0-kg masses at both ends.
1.0 kg
1.0 kg
Suppose it rotates about an axis through its center.
0.5 m
0.5 m
(a) Calculate the baton’s mass of inertia.
(b) Suppose a constant torque of 1.0 N-m is applied to the baton. What is its angular acceleration α ?
(c) If the baton starts from rest (ωi = 0 rad/s), what is its angular speed ω after t = 2 s ?
(d) Suppose, instead, the baton was shorter, but the torque was the same. Would ω at t = 2 s be larger or smaller ?

8. “Extended” Objects
For an “extended” solid object, such as a sphere, calculus techniques are used to calculate the object’s moment of inertia. (take PHY 231, 404, …) R
One simple example (which doesn’t require calculus) …
Consider a solid hoop of mass M and radius R. Think of it as being made up of many small masses m1, m2, m3, …, with M = Σm.
The moment of inertia due to all of these masses is: R m1 m2 m3

9. “Extended” Objects
Moments of inertia for some common “extended” objects :
Table 8.1 of your textbook
Don’t worry, you don’t need to “memorize” these formulas …
But you need to know how to apply/use them !

10. Example
A solid cylinder with radius m rotates on a frictionless, vertical axle. A constant tangential force of 250 N applied to its edge causes an angular acceleration of rad/s2.
(a) What is the moment of inertia ?
(b) What is the mass ?
(c) If the cylinder starts from rest, what is the angular speed after 5.00 s have elapsed (assuming the force is acting the entire time) ?

11. Example F
A 150-kg merry-go-round in the shape of a uniform, solid, horizontal disc of radius 1.50-m is set spinning by wrapping a rope about the rim of the disc, and pulling on the rope.
1.5 m
What constant force must be exerted on the rope to bring the merry-go-round from rest to an angular speed of 0.5 rev/s in a time of 2.0 s ?

12. Next Class 8.6 : Rotational Kinetic Energy
+ The “Race of the Shapes” !!