Rotational Motion Lecture Notes

Relating Linear and Angular Quantities

Clicker Question 3

• A vinyl record spins at a steady rate.
• Consider a small piece of the vinyl as it spins.
• What type of velocity does it have?
  • A) Only linear velocity, $\vec{v}$
  • B) Only angular velocity about the center, $\omega$
  • C) Both $\vec{v}$ and $\omega$
  • D) Neither $\vec{v}$ nor $\omega$

Basic Idea

• Length of track $\Delta x$ = arc length $\Delta s$
• Arc length is $R \Delta \theta$
• Therefore, Rolling Without Slipping
  • $v<em>{axel} = R</em>{wheel} \omega<em>{axel} = R</em>{wheel} \frac{\Delta \theta}{R_{wheel}}$
  • $\vec{v}_{axel}$: Speed of axel in frame of ground
  • $\omega_{axel}$: Angular speed of wheel about the axel
  • $R_{wheel}$: Radius of wheel

Clicker Question 4

• An old fashion bicycle travels down the road.
• Which bike wheel has the greater angular velocity about the axle?
  • A) The small wheel
  • B) The large wheel
  • C) They're the same
  • D) Not enough information

Velocity Acceleration Angular & Linear in General

• Velocity
  • $v_t = r \omega$: tangential component
  • $\omega$: Angular velocity
  • $r$: Distance from origin
• Acceleration
  • $a_t = r \alpha$: tangential component
  • $\alpha$: Angular acceleration
  • $r$: Distance from origin

Clicker Question 5

• A tree snaps and falls over, rotating about the point where it cracked.
• How does the angular speed of the leaves compare to the angular speed of the trunk?
• Assume the shape of the tree does not change as it falls.
  • A) Leaves have a higher angular speed
  • B) Trunk has a higher angular speed
  • C) They have the same angular speed

Clicker Question 6

• A tree snaps and falls over, rotating about the point where it cracked.
• How does the speed of the leaves compare to the speed of the trunk?
• Assume the shape of the tree does not change as it falls.
  • A) Leaves have a higher speed
  • B) Trunk has a higher speed
  • C) They have the same speed

Torque

What Causes Rotation?

• Force is required to speed up or slow down rotation
• Important factors
  • Magnitude of force
  • Direction of force
  • Location force is applied
• Overall result: $F_{tangent} r$
  • $F_{tangent}$: Tangential component of force
  • $r$: Distance from origin (pivot here)

Definition

• Sign convention
  • $\tau$ positive if counterclockwise twist
  • $\tau$ negative if clockwise twist
  • $r$ & $F_{\perp}$ both positive
• Notation notes
  • $\tau$ is lowercase Greek letter “tau”
  • $F<em>{\perp}$ also called $F</em>{tangent}$
• Torque, $\tau$
  • $\tau = \pm r F_{\perp}$
    • Torque
    • Distance from origin
    • Component of force perpendicular to $\vec{r}$

Equivalent definition

• Sign convention
  • $\tau$ positive if counterclockwise twist
  • $\tau$ negative if clockwise twist
  • $r$ & $F$ both positive
• Torque, $\tau$
  • $\tau = \pm rF \sin \theta$
    • Torque
    • Distance from origin
    • Magnitude of force
    • Angle between $\vec{r}$ & $\vec{F}$

Equivalent definition

• Sign convention
  • $\tau$ positive if counterclockwise twist
  • $\tau$ negative if clockwise twist
  • $r_{\perp}$ & $F$ both positive
• Notation notes
  • $r_{\perp}$ also called “moment arm”
  • $r_{\perp}$ is the closest distance to a line through $\vec{F}$
• Torque, $\tau$
  • $\tau = \pm r_{\perp} F$
    • Torque
    • Component of $\vec{r}$ perpendicular to $\vec{F}$
    • Magnitude of force

Clicker Question 7

• Two equal mass cats stand on tree limb.
• Cat A is twice is far from the trunk as cat B.
• How do the torques, about the base of the branch, on the branch compare?
  • A) $\tau<em>A > 2 \tau</em>B$
  • B) $\tau<em>A = 2 \tau</em>B$
  • C) $\tau<em>A = \tau</em>B$
  • D) $\tau<em>B = 2 \tau</em>A$
  • E) $\tau<em>B > 2 \tau</em>A$

Clicker Question 8

• Which force applies the most torque about the pin?
• (Forces to scale)

Center of Gravity

Basic idea

• Where gravitational force is applied for torque
• Same location as center of mass
  • Assuming uniform gravitational field
• Explanation
  • Gravitational force is distributed over an entire object
  • Often the gravitational force produces a torque
  • Acts as if all of force is applied at center of gravity

Clicker Question 9

• What is the torque on this uniform bar by its weight about the pivot point?
  • A) $mgL$
  • B) $mgx$
  • C) $mg (L-x)$
  • D) $-mg (L-x)$
  • E) $mg (\frac{L}{2} - x)$

Static Equilibrium

Basic idea

• Stationary implies conditions for force and torque
• Two conditions: Static Equilibrium Conditions
  • $\sum \tau = 0$: Net torque on object
  • $\sum \vec{F} = 0$: Net force on object
    • $\sum F_x = 0$
    • $\sum F_y = 0$
    • implies 3 equations in 2D

Clicker Question 10

• Two weights are placed on a seesaw, balancing it horizontally.
• Weight B is twice as massive as weight A.
• If A is a distance “x” from the pivot point, how far is weight B from the pivot point?
  • A) $2x$
  • B) $x$
  • C) $x/2$
  • D) Not enough information

Clicker Question 11

• A weight is placed on a 4 ft long wooden board, balancing it horizontally.
• The weight has the same mass as the board and the pivot is placed 1 ft from the end of the board.
• How far is the weight from the end of the board?
  • A) 0 ft
  • B) 0.5 ft
  • C) 1 ft
  • D) 2 ft
  • E) 3 ft

Problem Solving with Static Equilibrium

Choice of Origin

• Neat fact
  • If the net force on an object is zero, then the net torque is the same for any origin
• Choice of origin just makes the math easier or harder
  • Choose an origin where unknown or irrelevant forces produce no torques
  • Examples
    • At the location of an unknown force
    • At directly toward/away from an unknown force

Problem Solving Steps

1. Diagram the situation
   • Simplified picture with forces at location applied
   • Free body diagram can be helpful
2. Choose origin
   • Recommend location to reduce number of torques
   • Label distances to origin
3. Set up $\sum \tau = 0$
   • CCW torque +, CW torque -
4. Set up $\sum \vec{F} = 0$
   • Two components: $\sum F<em>x = 0$, $\sum F</em>y = 0$
5. Solve for variables of interest

Class Problem Simple Lever

• By pushing down perpendicularly on the lever, the operator manages to barely lift the box off the ground. If the lever is negligibly light…
  • Find the torque by the box on the lever
  • Find the torque by the hand on the lever
  • Find the magnitude of force applied by the hand

Class Problem Simple Lever Solution

• By pushing down perpendicularly on the lever, the operator manages to barely lift the box off the ground. If the lever is negligibly light…
  • Find the torque by the box on the lever: 24.5 Nm
  • Find the torque by the hand on the lever: -24.5 Nm
  • Find the magnitude of force applied by the hand: 66.9 N

Example Problem: Toppling Bench

• A 5 kg, 1.5 m long wooden plank is evenly set on top of two narrow posts 1 m apart.
• How much weight can be placed on one end before the plank flips over?
• What is the magnitude of force by each leg on the plank?

Example Problem: Toppling Bench Solution

• A 5 kg, 1.5 m long wooden plank is evenly set on top of two narrow posts 1 m apart.
• How much weight can be placed on one end before the plank flips over?
• What is the magnitude of force by each leg on the plank?
• Answer: M = 10 kg, FL = 147 N, FR = 0 N

Class Problem: Wire Tension

• A uniform 15 cm by 10 cm picture frame is hung from two wires.
  • What is the weight of the picture frame?
  • What is the tension in right wire?
  • What is the tension in the left wire?
  • What is the angle, $\theta$?

Class Problem: Wire Tension Solution

• A uniform 15 cm by 10 cm picture frame is hung from two wires.
  • What is the weight of the picture frame? 39.2 N
  • What is the tension in right wire? 29.4 N
  • What is the tension in the left wire? 49 N
  • What is the angle, $\theta$? 53.1°

Types of Equilibrium

Stable vs Unstable Equilibrium

• Basic idea
  • What happens if a static object is slightly bumped
• Stable equilibrium
  • Pushed back toward equilibrium
• Unstable equilibrium
  • Pushed away from equilibrium
• Neutral equilibrium
  • No force toward or away from equilibrium

Stability and Potential Energy

• Stable equilibrium
  • Potential energy maximum
• Unstable equilibrium
  • Potential energy minimum
• Neutral equilibrium
  • Constant potential energy

Clicker Question 12

• A round pencil rests on a level surface.
• What type of equilibrium is it in?
  • A) Stable equilibrium
  • B) Unstable equilibrium
  • C) Neutral equilibrium
  • D) Not in equilibrium

Clicker Question 13

• A marble rests at the bottom of a bowl.
• What type of equilibrium is it in?
  • A) Stable equilibrium
  • B) Unstable equilibrium
  • C) Neutral equilibrium
  • D) Not in equilibrium

Clicker Question 14

• A ball is momentarily at rest at the top of its upward trajectory.
• What type of equilibrium is it in?
  • A) Stable equilibrium
  • B) Unstable equilibrium
  • C) Neutral equilibrium
  • D) Not in equilibrium

Rotational Motion

Slides Overview

• Rigid Rotation about a Fixed Axis
  • Newton’s 2nd Law for Rotations
  • Moment of Inertia
• Energy and Rotations
  • Rotational Kinetic Energy
  • Work from Rotations
• Angular Momentum
  • Intro to Angular Momentum
  • Conservation of Angular Momentum

Rigid Rotation About a Fixed Axis

Rotations in 2D Plane vs About a Fixed Axis

• Many 3D situations can use 2D equations
  • Example: door on a hinge
• How?
  • Take “top down” perspective along axis
  • Ignore components along axis of rotation
  • Use distance from axis not origin
  • Component of force along axis irrelevant
• Requires pivot to prevent other rotation

Newton’s 2nd Laws for Rotation

• “Newton’s 2nd Law” for angular quantities
• Caveats
  • Requires rigid object rotating about a fixed objects
  • Ex: does not work for point mass moving in straight line
• $\sum \tau = I \alpha$
  • How you twist, what you twist, how it responds
  • Net torque on object, Moment of inertia, Angular acceleration
• $\sum \vec{F} = m \vec{a}$
  • How you push, what you push, how it responds

Example Problem: Spinning Wheels

• A freely spinning bike wheel is brought to a stop by contact with a brake pad at the outer edge.
• The wheel has a moment of inertia about the axel of 0.32 $kgm^2$ and a radius of 40 cm.
• If the coefficient of kinetic friction between the brake pad and wheel is 0.9 and the normal force applied to the wheel is 10 N, how long does it take for a wheel spinning at 80 rpm to come to a full stop?

Example Problem: Spinning Wheels Solution

• A freely spinning bike wheel is brought to a stop by contact with a brake pad at the outer edge.
• The wheel has a moment of inertia about the axel of 0.32 $kgm^2$ and a radius of 40 cm.
• If the coefficient of kinetic friction between the brake pad and wheel is 0.9 and the normal force applied to the wheel is 10 N, how long does it take for a wheel spinning at 80 rpm to come to a full stop?
• Answer: 0.745 s

Moment of Inertia

Moment of Inertia, I

• Basic idea
  • Resistance to rotation, or
  • How difficult it is to rotate an object back and forth
• More precisely
  • A particular combination of mass and distance from axis of rotation
• Note
  • Depends on shape
  • Depends on mass
  • Depends on axis of rotation
• $I = \sum \tau \alpha$

Calculating Moment of Inertia

• For a single point mass
  • $I = mr^2$
• For multiple point masses
  • $I = \sum<em>i m</em>i r_i^2$
• For continuous masses
  • Look it up in a table

Clicker Question 1

• Which object has the largest moment of inertia about this axis?
  • A) 1 kg at 1 m
  • B) 1 kg at 2 m
  • C) 2 kg at 1 m
  • D) A & C tie
  • E) All tie

Clicker Question 2

• Which object has the largest moment of inertia about this axis?
  • A) 2 kg at 1 m
  • B) 1 kg at 1 m (two of them)
  • C) 1 kg at 1 m, 1 kg at 1 m (different configuration)
  • D) 1 kg at 1 m, 1 kg at 1 m (another different configuration)
  • E) All tie

Class Problem: Moment of Inertia of Three Point Masses

• Find the moment of inertia of these three point-masses about the 3 kg mass.

Class Problem: Moment of Inertia of Three Point Masses Solution

• Find the moment of inertia of these three point-masses about the 3 kg mass.
• Answer: 23.25 $kgm^2$

Parallel Axis Theorem

• Basic idea
  • Moment of Inertia is smallest when the axis of rotation crosses the center of mass
  • For all other axes, compare to parallel axis through center of mass
• Parallel axis theorem
  • $I = I_{CM} + Md^2$
    • I : Moment of Inertia about the parallel axis
    • $I_{CM}$: Moment of Inertia about the center of mass
    • M: Total Mass
    • d: Distance between axes

Energy & Rotation

Rotational Kinetic Energy

Clicker Question 3

• Different wheels, with the same mass and radius, roll down an inclined plane.
• Wheel A is a solid disk and Wheel B is a hoop.
• Which one wins?
  • A) Wheel A
  • B) Wheel B
  • C) They Tie
  • D) Not Enough Info

Rotational Kinetic Energy

• Bike wheel demonstration Basic idea
  • There is kinetic energy even if the center of mass is stationary
• Depends on
  • Angular velocity ($\omega$)
    • Measured about axis of rotation
  • Moment of inertia ($I$)
    • Measured about axis of rotation
• $KE_{rotational} = \frac{1}{2} I \omega^2$

Mechanical kinetic energy is split between rotational and translational

• $KE<em>{mechanical} = \frac{1}{2} m</em>{tot} v<em>{CM}^2 + \frac{1}{2} I</em>{CM} \omega_{CM}^2$
  • $KE_{translational}$
  • $KE_{rotational}$
  • $I_{CM}$: Moment of Inertia about center of mass
  • $v_{CM}$: Speed of center of mass
  • $m_{tot}$: Total mass of object
  • $\omega_{CM}$: Angular speed about center of mass

Reference Frame Issues

• Kinetic energy depends on choice of reference frame
  • Frames moving relative to each other
  • Measure different amounts of mechanical energy
  • Formula works for any frame
• $KE<em>{mechanical} = \frac{1}{2} m</em>{tot} v<em>{CM}^2 + \frac{1}{2} I</em>{CM} \omega_{CM}^2$
  • Measured in any frame center of mass frame

Equivalent Formula with Pivot Point

• Basic idea
  • At any given moment, the motion of a rigid object can be described as purely a rotation about some pivot point.
  • Measuring about that pivot point yields
• $KE<em>{mechanical} = \frac{1}{2} I</em>{pivot} \omega_{pivot}^2$
  • Moment of Inertia about pivot point
  • Angular speed about pivot point

Example Equivalency: Rolling Disk

• STANDARD FORMULA: $KE<em>{mechanical} = \frac{1}{2} m</em>{tot} v<em>{axel}^2 + \frac{1}{2} I</em>{CM} \omega_{CM}^2$
• PIVOT FORMULA: $KE<em>{mechanical} = \frac{1}{2} I</em>{pivot} \omega_{pivot}^2$

Example Situation: Rolling an Object Down a Slope

• Energy is conserved
  • No work from slope
• What energies change?
  • $PE<em>g$ decreases: $mg \Delta y</em>{CM}$
  • $KE$ increases: $\frac{1}{2} m v<em>{CM}^2 + \frac{1}{2} I</em>{CM} \omega_{CM}^2$
• Conservation of energy would imply
  • Note: $\omega<em>{CM}^2 = \frac{v</em>{CM}^2}{R^2}$
  • $v_f = \sqrt{\frac{2gH}{(1 + \frac{I}{mR^2})}}$
• Solid disk: $\frac{I}{mR^2} = \frac{1}{2}$
• Hoop: $\frac{I}{mR^2} = 1$
• Solid disk is faster!

Example Problem Yo Yo Speed

• A Yo Yo is released from rest and spins as it descends down the string.
• It is made from three disks each with the same mass.
• The inner disk’s radius is half of the outer disks’ radius.
• How fast is its center of mass moving when it reaches the end of the 0.7 m long string?

Example Problem Yo Yo Speed Solution

• A Yo Yo is released from rest and spins as it descends down the string.
• It is made from three disks each with the same mass.
• The inner disk’s radius is half of the outer disks’ radius.
• How fast is its center of mass moving when it reaches the end of the 0.7 m long string?
• ANSWER: $v_{CM} = 2.34 m/s$

Work from Rotation

Torque and Work

• Basic idea
  • Pushing along the path of a circle is equivalent to applying a torque over an angle
• Work from rotations
  • Goes into $KE_{rot}$
  • $W<em>{rot} = \frac{1}{2} I</em>f \omega<em>f^2 - \frac{1}{2} I</em>i \omega_i^2$
  • $\tau \Delta \theta = F_{\perp} \Delta s$
  • $W_{rot} = \tau \Delta \theta$
    • (if constant torque)

Torque and Power

• Basic idea
  • Rate of energy transfer due to a torque
• $P_{rot} = \tau \omega$
  • Power: Rate at which energy is transferred into object by torque while rotating
  • $\omega$: Angular velocity, How quickly object is spinning
  • $\tau$: Torque on object

Clicker Question 4

• A string is wound around a wheel that spins on an axis.
• A constant tension pulls on the wheel, spinning it faster and faster.
• After 3 rotations with a constant tension, what is the work done on the wheel by the string?
  • A) $3RF_T$
  • B) $3 \pi RF_T$
  • C) $\pi RF_T$
  • D) $6 \pi RF_T$
  • E) None of the above

Class Problem Spinning It Up

• A string is pulled to spin up a disk with a frictionless, stationary axel in the middle.
• The string is wound around the edge of the initially stationary disk, and it completes three full rotations before the string completely unwinds and disconnects.
  • Find the torque on the disk from the string
  • Find the work done on the disk by the string
  • Find the final angular velocity of the disk

Class Problem Spinning It Up Solution

• A string is pulled to spin up a disk with a frictionless, stationary axel in the middle.
• The string is wound around the edge of the initially stationary disk, and it completes three full rotations before the string completely unwinds and disconnects.
  • Find the torque on the disk from the string: 1.8 Nm
  • Find the work done on the disk by the string: 33.9 J
  • Find the final angular velocity of the disk: 51.9 rad/s

Intro to Angular Momentum

Angular vs. Linear Quantities

Angular Momentum, L

• Basic idea
  • How hard it is to stop an object from rotating
• SI units
  • $kg \frac{m^2}{s}$
• It’s literally what makes the world go ‘round.
• $L = I \omega$
  • Angular momentum of an object about some axis
  • Moment of inertia of the same object about the same axis
  • Angular velocity of the same object about the same axis

Common Misunderstandings About Angular Momentum

• You have linear momentum OR angular momentum depending on if you rotate or move in a straight line (NOT TRUE)
  • Truth: You can have both at the same time
• An object needs to spin about its center of mass to have angular momentum (NOT TRUE)
  • Truth: Any object moving not directly toward or away from origin has angular momentum

Clicker Question 5

• Two people, A and B, are walking directly forward, as shown.
• Which person, if any, has an angular momentum about the pole?
  • A) Only A
  • B) Only B
  • C) Both A and B
  • D) Neither A nor B

Angular Momentum of a Point Mass

• Scenario
  • A point mass moving in a straight line at a constant velocity
• How much angular momentum?
• $L<em>{point} = \pm m v r</em>{\perp}$
  • Angular momentum of the point particle
  • Mass of the point particle
  • $r_{\perp}$: Component of 𝒓 perpendicular to 𝑣 (also closest distance to line through 𝑣)
  • Speed of the point particle
  • Sign matches 𝜔 (+ CCW, - CW)

Angular Momentum vs Rotational Kinetic Energy

• Angular momentum
  • How hard it is to stop rotating
• Rotational kinetic energy
  • How much damage it can do
• Kinetic energy & momentum
  • $L = I \omega$
  • $KE_{rot} = \frac{1}{2} I \omega^2$
  • $KE_{rot} = \frac{L^2}{2I}$

Conservation of Angular Momentum

Angular Impulse, Jang

• Basic idea
  • Torqueing an object over time
  • A transfer of angular momentum
• Notes
  • Rarely given a symbol
  • Angular momentum is transferred via torque
• If $\tau$ varies over time
  • Area under $\tau$ vs t curve or
  • $J<em>{ang} = \tau</em>{avg} \Delta t$
• $J_{ang} = \tau \Delta t$
  • (constant torque)
  • Angular impulse over some time interval
  • Torque during the time interval
  • Duration of time interval
  • (varying torque)

Torque & Angular Momentum

• Basic idea
  • Net torque is the rate of change in angular momentum over time
• Notes
  • Valid even when moment of inertia changes
  • Don’t need rigid rotation about an axis
• $\sum \tau = \frac{dL}{dt}$
  • Net torque
  • Slope of tangent line on angular momentum vs time graph

Conservation of Energy in Systems

• General equation
• How to use:
  • Choose a system
  • Choose a time interval
  • Replace $\Delta L_{tot}$ with sum of all angular momenta changes in system during time interval
  • Replace $\sum J_{ang, ext}$ with the sum of angular impulses by external objects on objects in system during time interval
• $\Delta L<em>{tot} = \sum J</em>{ang, ext}$
  • Total angular impulse from external torques
  • Change in total angular momentum of system

Clicker Question 6

• You're standing on a slick, spinning merry-go-round.
• You slide out to the edge and hold on.
• What happens to the angular velocity of the merry-go-round in this process?
  • A) It speeds up
  • B) It slows down
  • C) It stays the same
  • D) Not enough information

Clicker Question 7

• A small child runs up to the edge of a still merry-go-round and jumps on.
• They begin to spin, together.
• Is angular momentum, about the center of the merry-go-round, conserved?
  • A) Yes
  • B) No

Class Problem Merry Go Round

• A 30 kg child jumps onto the edge of an 80 kg merry-go-round at 2 m/s, as shown.
• What is the angular velocity of the merry-go-round as they rotate together afterward?

Class Problem Merry Go Round Solution

• A 30 kg child jumps onto the edge of an 80 kg merry-go-round at 2 m/s, as shown.
• What is the angular velocity of the merry-go-round as they rotate together afterward?
• ANSWER: 0.34 rad/s

Clicker Question 8

• Two astronauts are held together by a rope, but they are spinning.
• They pull closer to each other, which doesn’t change the total angular momentum.
• What happens to the rotational kinetic energy of the two astronauts?
  • A) Increases
  • B) Decreases
  • C) Stays the same

Example Problem Astronaut Rescue

• Two 60 kg astronauts are connected by a 10 m light tether.
• They spin rather slowly at a 0.1 rad/s angular velocity about their center of mass.
• One astronaut tries to reach the other by pulling on the tether until they are only 1 m apart.
  • What is their angular velocity when 1 m apart?
  • How much work would be required to bring them to this distance?
  • What is the tension in the rope at this distance?

Example Problem Astronaut Rescue Solution

• Two 60 kg astronauts are connected by a 10 m light tether.
• They spin rather slowly at a 0.1 rad/s angular velocity about their center of mass.
• One astronaut tries to reach the other by pulling on the tether until they are only 1 m apart.
  • What is their angular velocity when 1 m apart? 10 rad/s
  • How much work would be required to bring them to this distance? 1485 J
  • What is the tension in the rope at this distance? 3,000 N