Rotational Mechanics

AP Physics C: Mechanics

Enough with the particles…

• Do you ever get tired of being treated like a particle?

• We can not continue to lump all objects together and pretend that they undergo the same motion when acted upon by the same force…

• We will now study the rotation of rigid bodies.

What do we already know?

Objects rotate about their

center of mass.

For rotational motion it is useful to consider

tangential and radial

components instead of x

and yIdeas of circular

motion and centripetal

force.

What is a rigid body?• An extended object whose size

and shape do not change as it moves.

• This size and shape can not be neglected when modeling its motion.

• Objects that are held together by “massless rods” of molecular bonds.

• Some objects can be modeled as rigid during parts of their motion.

Three types of motion of a rigid

body:

There is an rotational analogy to

every concept of linear motion.• We have looked at the basics when studying

circular motion:

s r

ddt

ddt

vt ds

dt

a t dvt

dt

vt d r

dtr

a t d r

dtr

Tangential Components:

vt r

vt

r

a t

r

a t rω

vt

r

Rotational Velocity

Rotational Acceleration

Tangential Velocity

Tangential Acceleration

Radial (centripetal) Component

ar vt

2

r

ar r 2

r

vr 0

ar 2r

ω

vt Radial Velocity

Radial Acceleration

ar

Rotational Kinematic Equations:

1

2t 2 0t 0

0 t

2 02 2

v v0 at

d 1

2at 2 v0t d0

v2 v02 2ad

Sign Conventions:Counter-clockwise is positive direction

for ω and vt. So positive α can be speeding up in the ccw direction or slowing down in the cw direction.

A Rotating Crankshaft

A car engine is idling at 500rpm. When the light turns green, the crankshaft rotation speeds up at a constant rate

to 2500rpm over an interval of 3 seconds. How many revolutions does the crankshaft make during this time

interval?This is a rotating rigid body with constant angular

acceleration

Imagine painting a dot on the crankshaft. IF the dot is at θ=0 and t=0, at a

later time, the dot will be at:

1

2t 2 0t

A Rotating Crankshaft

A car engine is idling at 500rpm. When the light turns green, the crankshaft rotation speeds up at a constant rate

to 2500rpm over an interval of 3 seconds. How many revolutions does the crankshaft make during this time

interval?

0

t

0 500rev

min

1min

60sec

2rad

rev

52.4rad /s

2500rev

min50 262rad /s

A Rotating Crankshaft

A car engine is idling at 500rpm. When the light turns green, the crankshaft

rotation speeds up at a constant rate to 2500rpm over an interval of 3 seconds.

How many revolutions does the crankshaft make during this time interval?

1

2t 2 0t

1

269.9 32 52.4 3 472rad

0

t

262 52.4 rad /s

3s69.9rad /s2

472rad1rev

2rad

75revs

The Center of MassA 500g ball and a 2kg ball are

connected by a massless 50cm long rod.

Where is the center of mass?

What is the speed of each ball if they rotate about the center of mass at

40rpm?

The Center of MassA 500g ball and a 2kg ball are

connected by a massless 50cm long rod.

Where is the center of mass?

xcm 2kg 0 0.5kg 0.5

2.5kg0.10m

The Center of MassA 500g ball and a 2kg ball are

connected by a massless 50cm long rod.

What is the speed of each ball if they rotate about the center of mass at

40rpm?

xcm 0.10m

40rev

min

1min

60s

2rad

1rev

4.16rad /s

vt1 r1 0.10m 4.16rad /s 0.42m/s

vt 2 r2 0.40m 4.16rad /s 1.68m/s

What can we measure?

• What can we measure and analyze with a bike in order to further understand rotational mechanics?

Other Rotational Analogs?

Rotational Force?

Rotational Energy?

Rotational Momentum?

Rotational Mass?

Rotational Energy?

• Why does the Sun rise in the morning? Why do magnets stick together? Because everybody says so. Everybody.

• - Michael Scott

Rotational Energy?

Rotational EnergyWhy must there be such a thing?

K t 1

2mv2

All of the atoms in the object are moving so they must have kinetic energy!

Translational Kinetic Energy

Can we use our analogies to find an expression for Rotational

Kinetic Energy?

Rotational KE for a particle traveling in a

circle:

K t K r

v r

K t 1

2mv2

m ?

K r 1

2m r 2

K r 1

2mr22

m I

K r 1

2mr2 2

K r 1

2I2

Moment of Inertia• Has nothing to do with a moment in time. The

word comes from the Latin momentum which means motion.

• The rotational analog to mass.

• Describes the distribution of mass relative to the axis of rotation.

• Is different for each shape and orientation.

• An object with a large mass is hard to accelerate, an object with a large moment of inertia is difficult to rotate.

Moment of Inertia (Rotational Laziness)

• Inertia is the resistance to changes in motion

• Moment of inertia is the resistance to changes in rotation.

Moment of InertiaConsider an irregular shape that is rotating:

The object’s rotational energy is the sum of the kinetic energies of each

particle

K r 1

2m1 r 1

2 1

2m2 r 2

2 ...

K r 1

2miri

2 2

I miri2Moment of Inertia can be calculated as the

sum of the contributions from each particle in an object

Moment of Inertia

Calculating moment of

inertia can be very difficult for odd shapes. I for

many shapes has been

tabulated and printed in

handbooks for scientists and

engineers.

Note about the axis…

If the rotation axis is not through the center of mass, then rotation may cause the center of mass to move up

or down in a gravitational field. The gravitational potential energy of the object will change as it spins.

With no friction in the axle or other dissipative forces, the mechanical

energy can be described as:

E mech K r Ug 1

2I2 Mgycm

The three masses, held together by lightweight plastic rods, rotate about an axle passing through the right–

angle corner. At what angular velocity does the triangle have 100mJ of rotational energy?

150g

250g300g

axleω

6cm

8cm

I miri2

K r 1

2I2

I 0.15kg 0.06m 2 0.25kg 0.08m 2 0.3kg 0 2

I 2.14 10 3kgm2

The three masses, held together by lightweight plastic rods, rotate about an axle passing through the right–

angle corner. At what angular velocity does the triangle have 100mJ of rotational energy?

150g

250g300g

axleω

6cm

8cm

K r 1

2I2

2K r

I

I 2.14 10 3kgm2

2 100 10 3 J

2.14 10 3kgm2 9.67rad /s

9.67rad /s60s

1min

1rev

2

92rpm

A 1m long, 0.2kg rod is hinged at one end and connected to a wall. It is held out horizontally, then released. What is the speed of the tip of

the rod as it hits the wall?Conservation of

Energy:

E i E f

Mgycm 1

2I2

K r Ug i K r Ug

f

MgL

2

1

2

1

3ML2

2

g

2

L

62

A 1m long, 0.2kg rod is hinged at one end and connected to a wall. It is held out horizontally, then released. What is the speed of the tip of

the rod as it hits the wall?Conservation of

Energy:

g

2

L

62

3g

L

vt

r

r L

L3g

Lvt

vt 3gL 5.4m/s

Calculating Moment of Inertia:

Like finding inertia, we can not simply place the object on a scale to find its

moment of inertia. We must go through the calculation.

I ri2miRecall: Moment of Inertia can be calculated as

the sum of the contributions from each particle in an object

as Δm approaches zero it can be replaced with the differential dm.

I r2dm

Calculating Moment of Inertia: Tips

I r2dmBreak the object into elements that you will sum

together. Do this in a way that keeps the same distance from the axis for all particles in each element.

You will sum the elements over a range of distances so you must find an expression to substitute dm with a

differential dx, dy, or dz.

M

L

dm

dx

Densities are helpful but not necessarily the only way to solve:

M

A

dm

dA

M

V

dm

dVFor a complex object made up of parts with known

moments of inertia, sum the terms of each to find part to find the moment of inertia of the object:

Iobject I1 I2 I3 ...

Find the moment of inertia of a circular disk of radius R and mass M that rotates

on an axis passing through its center.

I r2dm0

R

M

A

dm

dA

A R2

dA 2rdr

I r2 M

R22rdr

0

R

Find the moment of inertia of a circular disk of radius R and mass M that rotates

on an axis passing through its center.

I r2 M

R2 2rdr0

R

I 2M

R2 r3dr0

R

I 2M

R2

r4

4

0

R

I MR2

2

The four T’s in the diagram are made from identical rods. Rank in order, from

largest to smallest, the moments of inertia for rotation about each dashed

line.

Ia>Id>Ib>I

c

Which has the most mass distributed farthest from the axis???

Parallel-Axis Theorem

We have been calculating the moment of inertia for rotational axes that run through the center of mass. This

theorem helps us if we wish to use an off-center axis but know where it is in relation to a parallel, on-center axis.

I Icm Md2

Parallel-Axis Theorem

Find the moment of inertia of a thin rod with mass M and length L about an axis

1/3rd of the length from one end.

I Icm Md2

Icm 1

12ML2 Moment of inertia through the

center of mass of a thin rod from table.

I ML2

12

M

L

6

2

ML2

12

ML2

36

ML2

9

cm1/3rd

d=L/6

Rotational Force?Why must there be such a thing?

Because all net forces do not cause rotation!

How can rotational force be maximized?

How do we calculate it?

Apply a force at the proper location.

Torque

• Torque is to rotational motion as force is to linear motion.

• Torque is given the Greek symbol capital tau (τ)

Torque• Is greater with a greater

force

• Is greater if the force is applied farther from the axis or rotation.

• Is greater if the angle of application of the force is perpendicular to the radial line.

F

r

sin

Frsin

F r

Torque

F r

Torque and Lever arm or Moment Arm

How much Torque does Luis Provide?

Rank the Torques

τe>τa=τd>τb

>τc

No Change in rotation

Analog to Newton’s 2nd Law

0No Change in

motion (no acceleration)

F 0

EQUILIBRIUM

Torque due to gravity

grav MgxcmCenter of mass

is relative to the axis of

rotation

Torque due to gravityThe gravitational torque is found by treating the object as if all its mass were concentrated at the center of

mass.

What is the torque on the 500kg steel

beam?

grav Mgxcm 500kg 9.8m/s2 0.8 3920Nm

Torque

Equilibrium Lab Challenge:

• Please DO NOT touch or alter any set-up.

• Report the unknown mass in each. I will collect one per group for score based on amount correct.

• 7 Minutes per table remaining time may be used to reconsider past tables but not revisit them while other groups are working.

• Bonus goes to group with most correct.

Rotational DynamicsA net centripetal force will cause an object’s path to change direction.

Fc mac mvt

2

r m2r

A net tangential force will cause a rotating object to speed up or slow

down.

Ft mat mr at ac

Rotational DynamicsOnly component of force tangent to

a circle causes rotation change:

Ft mat mr

at ac

F r Ftr

Ftr mr r mr2 I

Surprised?

Analog to Newton’s 2nd Law

I

F maUnbalanced

ForcesUnbalanced

Torques

A net torque causes an angular acceleration!

Rank the angular accelerations!

αb > αa > αc = αd = αe

Angular Momentum (L)

L I

p m

v

Linear Momentum Angular Momentum

m I

v

p

F t

L

t

L

F

r t

L

p

r

F

Angular Momentum for a particle in circular motion:

L I

mr2 mr2

v

r

mr

v

Angular Momentum

net

dL

dt

F net

dp

dt

Newton’s 2nd Law:

A net torque causes a particles angular

momentum to change.

The rate of change of a systems

angular momentum is the net torque on

the system.

Conservation of Angular Momentum:The angular momentum of an isolated (no

external torques) system is conserved.

L i

L f

A note on direction:Why is it easier to balance a bike

when the wheels are spinning than when they are not?

An object with a large linear momentum is difficult to slow down or to knock it off of its

straight line path.An object with a large angular momentum is

difficult to slow down or to change the direction of its axis

L

p

r

F

r

General Direction of Cross Product: The

RHR1. Point the fingers of your right hand in the direction of

vector A. 2. Close your fingers into the direction of vector B.3. Your thumb points in the direction of vector C.

Gyroscope!

PrecessionUse of a gyroscope

Gyrobowl!