Previous chapters described motion along a straight line› Translational (linear) motion

This chapter we will focus on rotational motion around some fixed axis.

Motion can be more fully described by both translational and rotational motion› Rolling

Rolling Motion

Rolling Motion

Two views of rolling motion: 1) Pure rotation aroundthe instantaneous axis or 2) rotation and translation.

Rotational Motion

Need similar concepts for objects moving in circle (CD, merry-go-round, etc.)

As before:› need a fixed reference system (line)› use polar coordinate system

Previously, we defined a displacement in translational motion as

Δx = x – x0

In order to define displacement for rotation we will use an angular measure called the radian

› Θ = 1 radian ≈ 57°

The radian can be defined as the arc length s along a circle divided by the radius r

Comparing degrees and radians

Converting from degrees to radians

3.5 7

2

3 6 0ra d1

]r e e s[d e g1 8 0

]r a d[

Angular Displacement

Every point on the object undergoes circular motion about the point O

Angles generally need to be measured in radians

Note:

r

s

3.5 72

3 6 01

r a d

]d e g r e e s[1 8 0

]r a d[

length of arclength of arc

radiusradius

Axis of rotation is the center of the disk› Fixed origin O

Need a fixed reference line› Usually, x-axis

During time t, the reference line moves through angle θ

Angular Displacement

The angular displacement is defined as the angle the object rotates through during some time interval

Every point on the disc undergoes the same angular displacement in any given time interval

if

Rigid Body› Every point on the object undergoes

circular motion about the point O› All parts of the object of the body rotate

through the same angle during the same time

› The object is considered to be a rigid body This means that each part of the body is

fixed in position relative to all other parts of the body

Translation MotionΔx = xf – xi

Rotational MotionΔθ = θf - θi

The unit of angular displacement is the

radian Each point on the object undergoes the same

angular displacement

Angular Velocity

The average angular velocity (speed), ω, of a rotating rigid object is the ratio of the angular displacement to the time interval

Analogous to linear motion

ttt if

if

The instantaneous angular speed is defined as the limit of the average speed as the time interval approaches zero

Units of angular speed are radians/sec› rad/s

Speed will be positive if θ is increasing (counterclockwise)

Speed will be negative if θ is decreasing (clockwise)

Angular Acceleration

What if object is initially at rest and then begins to rotate?

The average angular acceleration, , of an object is defined as the ratio of the change in the angular speed to the time it takes for the object to undergo the change:

Units are rad/s² Similarly, instant. angular accel.:

ttt if

if

tt

0

lim

Angular Acceleration Units of angular acceleration are rad/s² Every point of the object has the same angular

speed and the same angular acceleration The sign of the acceleration does not have to be

the same as the sign of the angular speed Angular acceleration is positive if an object

rotating counterclockwise is speeding up or if an object rotating clockwise is slowing down.

The instantaneous angular acceleration is defined as the limit of the average acceleration as the time interval approaches zero

Since rotational motion is analogous to translational motion we can apply equations of motion similar to linear motion with constant acceleration

Rotational Kinematics

Notes about angular kinematics:

When a rigid object rotates about a fixed axis, every portion of the object has the same angular speed and the same angular acceleration

i.e. , and are not dependent upon r, distance form hub or axis of rotation

Example: A wheel rotates with a constant angular acceleration of 3.50 rad/s2. If the angular velocity of the wheel is 2.00 rad/s at t=0,

a)Through what angle does the wheel rotate between t=0 and 2.00 s? (in radians and revolutions)

b)What is the angular velocity of the wheel at t = 2.00s?

Want to consider how angular quantities relate to linear quantities

Consider an arbitrarily shaped object

Recall s = rθ Determine change in

θ with respect to time

Starting with,Δθ = Δs / r

Divide by Δt, we arrive at

vt = rω

Where vt is the tangential speed

Similarly, applying the same procedure as before to the tangential speed, we get

at = rα

where at is the tangential acceleration

Angular quantities vs. Linear quantities on a rotating object

Every point on the rotating object has the same angular motionSame for all r

Every point on the rotating object does NOT have the same linear motionIncreases with increasing r

Relationship Between Angular and Linear Quantities Displacements

Speeds

Accelerations

rs

ConcepTest A ladybug sits at the outer edge of a merry-go-round, and

a gentleman bug sits halfway between her and the axis of rotation. The merry-go-round makes a complete revolution once each second.The gentleman bug’s angular speed is

1. half the ladybug’s.2. the same as the ladybug’s.3. twice the ladybug’s.4. impossible to determine

ConcepTest 2A ladybug sits at the outer edge of a merry-go-round, and

a gentleman bug sits halfway between her and the axis of rotation. The merry-go-round makes a complete revolution once each second.The gentleman bug’s angular speed is

1. half the ladybug’s.2. the same as the ladybug’s.3. twice the ladybug’s.4. impossible to determine

Note: both insects have an angular speed of 1 rev/s

The carnival ride, The Gravitron, spins at 40 revolutions per minute and has a radius of 3 meters.  If the distance traveled in one rotation is the circumference of the ride, what is your tangential (linear) speed if you are sitting on the outside edge of this ride?

Vector Nature of Angular Quantities

As in the linear case, displacement, velocity and acceleration are vectors:

Assign a positive or negative direction

A more complete way is by using the right hand rule› Grasp the axis of

rotation with your right hand

› Wrap your fingers in the direction of rotation

› Your thumb points in the direction of ω

Angular quantities as vectors› Instinctively, we

expect that something be moving along the direction of a vector

› This is NOT the case for angular quantities

› Instead, something is rotating around the direction of the vector

› In the world of rotation, a vector defines an axis of rotation

Velocity Directions› In a, the disk rotates

clockwise, the velocity is into the page

› In b, the disk rotates counterclockwise, the velocity is out of the page

Acceleration Directions› If the angular acceleration and the angular

velocity are in the same direction, the angular speed will increase with time

› If the angular acceleration and the angular velocity are in opposite directions, the angular speed will decrease with time

Dynamics of a Rigid Body› Previous chapters discussed whether a

force was applied or not, not where it was applied

› We can generalize concepts of force to describe rotational dynamics Generalize Newton’s Laws Equilibrium

› Extend conservation laws Energy Momentum

Torque

Consider force required to open door. Is it easier to open the door by pushing/pulling away from hinge or close to hinge?

closeclose to hinge to hinge

awayaway from from hingehinge

Farther Farther from from from from hinge, hinge, larger larger rotational rotational effect!effect!

Torque Consider a door

that rotates about a hinge› The door is free to

rotate about an axis through O

› Force is perpendicular

› τ = rF› + CCW

- CW

If the applied force is not perpendicular we must take the components of the Force vector

However, only a net torque perpendicular will cause it to rotate

The magnitude of the torque t exerted by the force F is

τ = rFsinθ

sin 90 = 1, perpendicular max force sin 270 = -1, negative dir. max force sin 0 = sin 180 = 0, no force

The value of τ depends on the chosen axis of rotation

› Once the axis is chosen, apply right hand rule to determine direction

1. Point fingers in direction of r2. Curl fingers in the direction of F3. Your thumb points in the direction of the

torqueTorque is out of the screen

Newton’s Law Analog1. The rate of an object does not change, unless

acted on by a net torque2. The angular acceleration of an object is

proportional to the net torque

There are three factors that determine the effectiveness of torque:

› The magnitude of the force› The position of the application of the force› The angle at which the force is applied

If the net torque is zero, the object’s rate of rotation doesn’t change

Equilibrium – arbitrary axis› May have convenient location› When solving a problem, you must

specify an axis of rotation and maintain it

ConcepTestYou are using a wrench and trying to loosen a rusty nut. Which of the arrangements shown is most effective in loosening the nut? List in order of descending efficiency the following arrangements:

ConcepTestYou are using a wrench and trying to loosen a rusty nut. Which of the arrangements shown is most effective in loosening the nut? List in order of descending efficiency the following arrangements:

2, 1, 4, 3 or2, 4, 1, 3

Example:

Net Torque

The net torque is the sum of all the torques produced by all the forces

› Remember to account for the direction of the tendency for rotation Counterclockwise torques are positive Clockwise torques are negative

What if two or more different forces act on lever arm?

Example 1:

Given:

weights: w1= 500 Nw2 = 800 N

lever arms: d1=4 m d2=2 m

Find:

= ?

1. Draw all applicable forces

(500 )(4 ) ( )(800 )(2 )

2000 1600

400

N m N m

N m N m

N m

2. Consider CCW rotation to be positive

500 N 800 N

4 m 2 m

Rotation would be CCW

N

Determine the net torque:

Torque and Equilibrium

First Condition of Equilibrium The net external force must be zero

› This is a necessary, but not sufficient, condition to ensure that an object is in complete mechanical equilibrium

› This is a statement of translational equilibrium Second Condition of Equilibrium

The net external torque must be zero

This is a statement of rotational equilibrium

0

0 an d 0x y

F

F F

VVVVVVVVVVVVVV

0

Axis of Rotation

So far we have chosen obvious axis of rotation If the object is in equilibrium, it does not

matter where you put the axis of rotation for calculating the net torque› The location of the axis of rotation is completely

arbitrary› Often the nature of the problem will suggest a

convenient location for the axis› When solving a problem, you must specify an

axis of rotation Once you have chosen an axis, you must maintain

that choice consistently throughout the problem

Equilibrium, once again

A zero net torque does not mean the absence of rotational motion› An object that rotates at uniform angular

velocity can be under the influence of a zero net torque This is analogous to the translational

situation where a zero net force does not mean the object is not in motion

Example of aFree Body Diagram

Isolate the object to be analyzed

Draw the free body diagram for that object› Include all the

external forces acting on the object

Where would the 500 N person have to be relative to fulcrum for zero torque?

Example:

Given:

weights: w1= 500 Nw2 = 800 N

lever arms: d1=4 m = 0

Find:

d2 = ?

1. Draw all applicable forces and moment arms

2

2 2

(800 )(2 )

(500 )( )

800 2 [ ] 500 [ ] 0 3.2

RHS

LHS

N m

N d m

N m d N m d m

500 N 800 N

2 md2 m

According to our understanding of torque there would be no rotation and no motion!

N’ y

What does it say about acceleration and force?

Thus, according to 2nd Newton’s law F=0 and a=0!

So far: net torque was zero.What if it is not?

Torque, , is the tendency of a force to rotate an object about some axis

Forces cause accelerationa = F/m = Δv / Δt

Torques cause angular accelerations› Angular acceleration about some fixed point O

at some length r

Force and torque must be related in some way

Torque and Angular Acceleration

When a rigid object is subject to a net torque (≠0), it undergoes an angular acceleration

The angular acceleration is directly proportional to the net torque› The relationship is

analogous to ∑F = ma Newton’s Second Law

Newton’s Second Law for a Rotating Object

The angular acceleration is directly proportional to the net torque

The angular acceleration is inversely proportional to the moment of inertia of the object

There is a major difference between moment of inertia and mass: the moment of inertia depends on the quantity of matter and its distribution in the rigid object.

The moment of inertia also depends upon the location of the axis of rotation

I

Moment of Inertia is rotational equivalent of mass

Objects with larger mass (more inertia) are harder to accelerate, objects with larger moments of inertia are harder to rotate› Easier to spin a wheel or rod with mass located

at center Objects moment of inertia depends on not

only on the object’s mass but how the mass is distributed

Moment of Inertia for particles

I = Σmr2 = m1r12+ m2r2

2+m3r32

r2

r1

r3

m1

m2

m3

Moment of Inertia of a Uniform Ring

• Calculate the moment of inertia

• Imagine the hoop is divided into a number of small segments, m1 …

• These segments are equidistant from the axis 2 2

i iI m r M R

Other moments of inertia

The rotational inertia of an object depends not only on its mass distribution but also the location of the axis of rotation – compare (f) and (g), for example.

a) a) solid aluminum

b) hollow goldb) hollow gold

c) samec) same

same mass & radius

solid hollow

Two spheres have the same radius and equal masses. One is made of solid aluminum, and the other is made from a hollow shell of gold.

Which one has the bigger moment of inertia about an axis through its center?

a) a) solid aluminum

b) hollow goldb) hollow gold

c) samec) same

same mass & radius

solid hollow

Two spheres have the same radius and equal masses. One is made of solid aluminum, and the other is made from a hollow shell of gold.

Which one has the bigger moment of inertia about an axis through its center?

Moment of inertia depends on mass and distance from axis squared. It is bigger for the shell since its mass is located farther from the center.

Total Energy of Rotating System An object rotating about some axis with an angular

speed, ω, has rotational kinetic energy ½Iω2

Energy concepts can be useful for simplifying the analysis of rotational motion

Conservation of Mechanical Energy

› Remember, this is for conservative forces, no dissipative forces such as friction can be present

› Rolling race!

fgrtigrt P EK EK EP EK EK E )()(

Rotational Kinetic EnergyWhen using conservation of energy, both rotational and translational kinetic energy must be taken into account.All these objects have the same potential energy at the top, but the time it takes them to get down the incline depends on how much rotational inertia they have.

Other moments of inertia

The rotational inertia of an object depends not only on its mass distribution but also the location of the axis of rotation – compare (f) and (g), for example.

Work-Energy in a Rotating System In the case where there are dissipative

forces such as friction, use the generalized Work-Energy Theorem instead of Conservation of Energy

Wnc = KEt + KER + PE

Note on problem solving:

The same basic techniques that were used in linear motion can be applied to rotational motion.› Analogies: F becomes , m becomes I

and a becomes , v becomes ω and x becomes θ

Techniques for conservation of energy are the same as for linear systems, as long as you include the rotational kinetic energy

Example: Problem Use conservation of energy to determine the angular speed of the spool after the bucket has fallen 4.00m starting from rest. The light string attached to the bucket is wrapped around the spool and does not slip as it unwinds.

Similarly to the relationship between force and momentum in a linear system, we can show the relationship between torque and angular momentum

Angular momentum is defined as L = I ω

› and Lt

Angular Momentum

Similarly to the relationship between force and momentum in a linear system, we can show the relationship between torque and angular momentum

Angular momentum is defined as L = I ω

If the net torque is zero, the angular momentum remains constant

Conservation of Linear Momentum states: The angular momentum of a system is conserved when the net external torque acting on the systems is zero.

› That is, when

t

L

ffiifi IIo rLL ,0

t

pF

(compare to )

In an isolated system, the following quantities are conserved:› Mechanical energy› Linear momentum› Angular momentum

With hands and feet drawn closer to the body, the skater’s angular speed increases› L is conserved, I

decreases, increases

› Ice skater

Example: