1. Aim: How do we apply Newton’s 2nd Law of Rotational Motion?

2. Newton’s 2nd Law of Rotational Motion
The net torque on a rigid body about an axis is equal to the rotational inertia of that body about the axis multiplied by the angular acceleration of the rigid body. τ=Iα

3. Torque Problem 1
Consider the figure below. Assume the mass of the bar is m and the length of the bar is L.
Find the initial torque produced by gravity.
τ=rF=mgL/2
b)Find the initial angular acceleration of the bar.
τ=Iα mgL/2=(1/3)mL2α α=3g/2L
c)Find the initial tangential acceleration of the center of the bar.
a=rα=(L/2)(3g/2L)=3/4 g
d)Find tangential acceleration of the right end of the bar.
a=rα=L(3g/2L)=3/2 g

4. Torque Problem 2 τ=rF=1.5(250)=375 N m
Consider the father pushing a playground merry-go-round in the figure. He exerts a force of 250 N at the edge of the kg merry-go-round, which has a 1.50-m radius. Calculate the angular acceleration produced (a) when no one is on the merry-go-round and (b) when an 18.0-kg child sits 1.25 m away from the center. Consider the merry-go-round itself to be a uniform disk with negligible friction.
τ=rF=1.5(250)=375 N m
I=1/2 MR2=1/2 (200)(1.5)2=225kg m2
Τ=Iα
375=225α
α=1.67 rad/s2
b) I=Imerry go round +Ichild
I=225 +mr2=225+18(1.25)2=225+28=253 kg m2
375=253α
α=1.48 rad/s2

5. Rigid Body under a net torque (Problem 3)
A model air plane with mass kg is tethered by a wire so that it flies in a circle 30 m in radius. The airplane engine provides a net thrust of N perpendicular to the tethering wire.
Find the torque the net thrust produces about the center of the circle.
ԏ=rFsinΘ=.8(30)sin90=2.4 N m
b) Find the angular acceleration of the airplane when it is in level flight.
I=mr2=.75(30)2 = .75(900)=675 kg m2
ԏ=Iα
2.4=675α
α= rad/s2
c) Find the tangential acceleration of the plane.
a=rα
a=30(.0356) = 1.07 m/s2
a) 24 N m b) rad/s^2 c)1.07 m/s^2

6. Rigid body under a net torque (Problem 4)
An electric motor turns a flywheel through a drive belt that joins a pulley on the motor and a pulley that is rigidly attached to the flywheel. The flywheel is a solid disk with a mass of 80.0 kg and a diameter of 1.25 m. It turns on a frictionless axle. Its pulley has much smaller mass and a radius of m. If the tension in the upper (taut) segment of the belt is 135 N and the flywheel has a clockwise angular acceleration of 1.67 rad/s2, find the tension in the lower (slack) segment of the belt.
21.5 N

7. Calculating Net Torque Problem 4

9. Problem 5-Atwood Machine with Massive Pulley
We have analyzed an Atwood machine in which two objects with unequal masses hang from a string that passes over a light, frictionless pulley. Suppose the pulley, which is modeled as a disk, has mass M and radius R, and the pulley surface is not frictionless, so that the string does not slide on the pulley. We will assume that the torque acting at the bearing of the pulley is negligible. Find the acceleration of the masses.