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 email contact this week. Send email with any questions and/or concerns. His office hours are cancelled this week.

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.

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”

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 !! ]

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”).

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.

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 ?

“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

“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 !

Example A solid cylinder with radius 0.330 m rotates on a frictionless, vertical axle. A constant tangential force of 250 N applied to its edge causes an angular acceleration of 0.940 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) ?

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 ?

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