Rotational Motion

Rotation about a fixed axis

Rotational Motion

Translation

Rotation

Rolling

Angular Kinematics

Describing Angular Motion

Angular Position -

Displacement is now defined as angle of rotation.

Polar coordinates (r, ) will make it easier to notate this motion.

Angular Position -

For a circle: =circumference/radius So: = s/r or s = r

s => arc length r => radius => angle, measured in radians

Remember: 360º = 2radians

Displacement:

–

Angular Velocity - Average angular

velocity

Instantaneous angular velocity

t

dt

d

Angular Direction

Right-hand coordinate system is positive when is increasing, which

happens in the counter-clockwise direction

is negative when is decreasing, which happens in the clockwise direction

Angular Acceleration - Average Angular

Acceleration

Instantaneous Angular Acceleration

t

dt

d

example 1

= (t3 - 27t + 4)rad At what time(s) does = 0

rad/s?

example 2

= (5t3 – 4t)rad/s2

At t = 0s: = 5 rad/s and = 2 rad.

Linear to Angular Conversions

Position: s = rVelocity: v = rAcceleration: a = r

Remember, also, centripetal accleration: ac = v2/r =2r

Angular Kinematics*constant

2

2

1

20

21

20

01

tt

t

example 3

A turntable starts from rest and begins rotating in a clockwise direction. 10 seconds later, it is rotating at 33.3 revolutions per minute.

What is the final angular velocity (rad/s)? What is the average angular acceleration? How far did a point 10 cm from the center

travel in that 10 seconds, both angularly and linearly?

example 4

In an astronaut training centrifuge (r = 15m):

What constant would give 11g’s?How fast is this in terms of linear speed?What is the translational acceleration to

get to this speed from rest in 2 minutes?

Moment of Inertia

“Rotational Mass”

Moment of Inertia

Description of the distribution of mass Measure of an object’s ability to resist a

change in rotation

Note: r is the perpendicular distance from the particle to the axis of rotation.

2i iI m r

example

Two children (m=40 kg) are on the teacup ride at King’s Dominion. The childrens’ center of mass is 0.75 m from the middle of the cup. What is the moment of inertia for children in the teacup?

Moment of Inertia forContinuous Mass Distribution

2

2

0

2

limi

i i

i im

I m r

I m r

I r dm

V

mdm dV

dm dA

dm dr

Calculate I for a hoop

A uniform hoop: mass (M); radius (R)

2

2

2

2

1

I r dm

I R dm

I R dm

I MR

Calculate I for a thin rod

A uniform thin rod: mass (M); length (L); rotating about its center of mass.

2

2

22

2

23

2

23

2

2

1

3

1

3

1

12

L

L

L

L

L

L

I r dm

I r dr

I r dr

I r

MI r

L

I ML

Common Moments of Inertia

Parallel Axis Theorem

To calculate the moment of inertia around any axis….

Where d is the distance between the center of mass and the axis of rotation

2cmI I Md

Calculate I for a thin rod rotating around the end

A uniform thin rod: mass (M); length (L); rotating about one end.

Rotational Energy

The kinetic energy an object has due to its rotational velocity.

Rotational Kinetic Energy

example

If you roll a disk and a hoop (of the same mass) down a ramp, which will win?

Torque

The tendency of a force to cause angular motion

Torque

Torque is dependent on the amount and location of the force applied to an object.

sin

r F

rF

Where r is the distance between the pivot point and the force and is the angle between r and F.

example 5

A one piece cylinder has a core section that protrudes from a larger drum. A rope wrapped around the large drum of radius, R, exerts a force, F1, to the right, while a rope wrapped around the core, radius r, exerts a force, F2 downward..

Calculate the net torque, in variables. If F1=5 N, R = 1 m, F2=6 N, and r = 0.5 m,

calculate the net torque.

Cross Product

The “other” vector multiplication

Cross product

Results in a vector quantity Calculates the perpendicular product of two

vectors The product of any two parallel vectors will

always be zero.

0ˆˆ

0ˆˆ

0ˆˆ

kk

jj

ii

The Right Hand Rule

Point your fingers along the radius

Curl them in the direction of the force

Your thumb will be pointing in the direction of the rotation.

http://hyperphysics.phy-astr.gsu.edu/hbase/tord.html

Rotational Direction is defined by the axis that it rotates around.

Cross Product

kijik

jkikj

ijkji

ˆˆˆˆˆ

ˆˆˆˆˆ

ˆˆˆˆˆ

Calculating Cross Product

kBkAjBkAiBkA

kBjAjBjAiBjA

kBiAjBiAiBiABA

kBjBiBkAjAiABA

zzyzxz

zyyyxy

zxyxxx

zyxzyx

ˆˆˆˆˆˆ

ˆˆˆˆˆˆ

ˆˆˆˆˆˆ

ˆˆˆˆˆˆ

Finding the Determinant

kFrFrjFrFriFrFrFr

kFF

rrj

FF

rri

FF

rrFr

FFF

rrr

kji

Fr

xyyxxzzxyzzy

yx

yx

zx

zx

zy

zy

zyx

zyx

ˆˆˆ

ˆˆˆ

ˆˆˆ

example 6

Find the cross product of 7i + 5j – 3k and 2i-8j + 7k.

What is the angle between these two vectors?

example 7

A plumber slips a piece of scrap pipe over his wrench handle to help loosing a pipe fitting. He then applies his full weight (900 N) to the end of the pipe by standing on it. The distance from the fitting to his foot is 0.8 m, and the wrench and pipe make a 19º angle with the ground. Find the magnitude and direction of the torque being applied.

example 7 continued…

r = 0.8m

F = 900 N19º

Nm

rF

680

)71sin()900)(8.0(

sin

example 8

One force acting on a machine part is F = (-5i + 4j)N. The vector from the origin to the point applied is r = (-0.5i + 0.2j)m.

Sketch r and F with respect to the origin Determine the direction of the force with the

right hand rule. Calculate the torque produced by this force. Verify that your direction agrees with your

calculation.

Equilibrium

Conditions for Equilibrium

1. Net force in all directions equals zero

2. Net torque about any point equals zero

0F

0

example 9

A 2 kg seesaw has one child (30 kg) sitting 2.5 from the pivot. At what distance from the pivot should a 25 kg child sit to balance the seesaw?

example 10

Find Tcable

example 11

Find T1 and T2

example 12

Find

Newton’s Second Law

When the sum of the torques does not equal zero

Newton’s Second Law

2

( )

( )

( )

F ma

F r

ma r

m r r

mr

I

example 13

Fpivot

Fnormal

Fgravity

2

2

1

2 3

3/

2

g

I

LF mL

grad s

L

L

example 14

Pulley – M1, R

Block – M2

Find a and FT

FT

Fg

2

2 2

2 1 2

2 2 1

2 2 1

2

2 1

1

2

1

21

2

12

y y

g T y

T y

y y

y y

y

y

F ma

F F M a

M g F M a

M g M a M a

M g M a M a

M g M M a

M ga

M M

2

1

1

21

2 1

1 2

2 1

1

2

1

2

1122

2

T

T

T

T

I

aF R M R

R

F M a

M gF M

M M

M M gF

M M

Angular Momentum

For a particle

Remember linear momentum:

So angular momentum:

mvp

sinrmvL

prL

Angular momentum and Torque

dt

dLt

Lt

mvrFr

t

mv

t

pF

For a system…

IL

rmL

rmrrmL

rv

rvmL

ii

iiiii

iii

2

2)(

To check…

I

Idt

d

dt

dL

IL

Conservation

If net torque is zero, then the angular momentum is constant.

0dt

dL

Conservation

For rotation about a fixed axis, we can say:

1100

10

II

LL

example

A star with a radius of 10,000 km rotates about its axis with a period of 30 days. If it undergoes a supernova explosion and collapses into a neutron star with a radius of 3 km, what is its new period?