Sample Demonstrations and Problems Involving Rotational

and Circular Motion

Car Rounding CurveCar Rounding Curve

What is the maximum speed in which a 2400 kg car can round a flat curve with a radius of 40 m if the friction coefficient between the tires and the dry road surface is 0.60?

How does this maximum speed change if the road is wet and the friction coefficient decreases to 0.20 (= 1/3 as much)?

How does this maximum speed change if the car has twice the mass (= 4800 kg)?

Ffriction = Fcentripetal

N = mv2/rmg = mv2/r v2 = gr

What are the implications of this equation?

Loop-the-LoopLoop-the-Loop

A loop-the-loop track consists of an incline that leads into a circular loop of radius r. What is the minimum height that a mass can be released from rest and still make it around the loop without falling off? Neglect friction.

At top of loop:

mv2/r = mgv2 = gr

Conservation of Energy:

mgh at the top = mgh + ½mv2 at the top of loopmgh = mg(2r) + ½mgr h = 2r + ½r 2h = 4r + r = 5r h = 5/2 r = 2.5r

What are the implications of this equation?What if we include friction?

What if the object rolled down the track?

Net Torque and Angular AccelerationNet Torque and Angular Acceleration

A mass m is attached to a light string that is wound around a cylindrical disk of radius R and mass M. What is the linear acceleration of the mass after it is released to fall to the floor below?

How long will it take to reach the floor a distance h below? How fast is it going when it reaches the floor?

10.5M m

m since g, a

g0.5Mm

mg

M 2m

2m a 2mg 2m) a(M

2mg 2ma Ma mamg2

Ma

a), - m(g T T, - mg ma :m mass hanging For the2

Ma T

M

2T

MR

2TRRαa

MR

2T

0.5MR

TR

Iα

2

τ

Once you know the linear acceleration of the mass, you solve for the time and speed in the usual manner.

Objects Rolling Down InclinesObjects Rolling Down Inclines

An object (e.g., solid cylinder, solid sphere, hoop, etc…) with mass m and radius r is positioned at the top of an incline. What is its speed when it reaches the bottom of the incline?

Conservation of Energy:Gravitational PE at the top = (KE + Rotational KE) at the bottom

mgh = ½mv2 + ½I2

mgh = ½mv2 + ½(substitute formula)(v/r)2

mgh = ½mv2 + ½(kmr2)(v2/r2) 2gh = v2 + kv2

What are the important implications of this equation?

Rotational Inertia and Angular Momentum DemosRotational Inertia and Angular Momentum Demos

• Act like a person who is losing their balance on an icy sidewalk and about to fall. Why do we do what we naturally do?• Try to balance a variety of objects on your finger • Balance baseball bat – which end is easier? Why?• Stick pencil/pen/dowel through rubber stopper and try to balance • What does an ice skater do when they are entering a spin? Why?• Twist meter sticks or rods with weights on each side• Stand or sit and spin on chair/stool/platform holding weights• Roll similar objects down incline• Rotating bicycle wheel