• Animation of what a Neutron Star Collision would look like.
• A pulsar is a rapidly rotating neutron star that emits radio waves in a beam that rotates with the star.

Jocelyn Bell Burnell

• This is a photo of the “scintillation array” used by Bell Burnell to discover PSR B1919+21.
• It was in a big field just outside Cambridge, UK, covered two acres and was made up of 1000s of posts and 120 miles of cable.
• She spent the first 2 years of her PhD building it. She did all the data analysis, discovered the pulsar, and wrote a famous paper about it.
• As a direct result, her thesis supervisor, Antony Hewish, won the Nobel Prize in 1974 for “his decisive role in the discovery of pulsars.”
• His “decisive role” was having Bell Burnell as a graduate student!

Suppose a horizontal disk is rotating on a lab bench and you are looking down on it.

The rotation axis passes through the centre, and is perpendicular to the disk (out of page)

Rotational kinematics

• There are similarities between the motions of different points on a rotating rigid body.
• During a particular time interval, all coins at the different points on the rotating disk turn through the same angle.
• Perhaps we should describe the rotational position of a rigid body using an angle.

Rotational (Angular) Position

Angular Velocity

Angular Acceleration

• Angular velocity, ω, is the rate of change of angular position, θ.
• The units of ω are rad/s.
• If the rotation is speeding up or slowing down, then its angular acceleration, α, is the rate of change of angular velocity, ω.
• The units of α are rad/s².
• All points on a rotating rigid body have the same ω and the same α.

From the Preclass Survey

Rotational Kinematics

Rotational Kinematics

• θ is angular position. The S.I. Unit is radians, where 2π radians = 360°.

• x specifies position. The S.I. Unit is metres.

Linear

Rotational Analogy

• Velocity, v_x, is the slope of the x vs t graph. [m/s]

• Acceleration, a_x, is the slope of the v_x vs t graph. [m/s²]

• Angular velocity, ω, is the slope of the θ vs t graph. [rad/s]

• Angular Acceleration, α, is the slope of the ω vs t graph. [rad/s²]

Radians are the Magical Unit!

• Radians appear and disappear as they please in your equations!!!
• They are the only unit that is allowed to do this!
• Example:

1 rad

Rotational Kinematics

Table 10.2

From the Preclass Survey

• Can you explain what is the unit rpm and rev in the textbook example 10.6 ?

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From today’s Preclass Survey

A bicycle wheel has an initial angular velocity of 1.50 rad/s, and a constant angular acceleration of 0.200 rad/s². Through what angle has the wheel turned between t = 0 and t = 2.50 s?

The “Rolling Without Skidding” Constraints

When a round object rolls without skidding, the distance the axis, or centre of mass, travels is equal to the change in angular position times the radius of the object.

d = θR

The speed of the centre of mass is

v = ωR

The acceleration of the centre of mass is

a = αR

Rotational Inertia

Depends upon:

• mass of object.
• distribution of mass around axis of rotation.
• The greater the distance between an object’s mass concentration and the axis, the greater the rotational inertia.

Rotational Inertia

Consider a body made of N particles, each of mass m_i, where i = 1 to N. Each particle is located a distance r_i from the axis of rotation. For this body made of a countable number of particles, the rotational inertia is:

The units of rotational inertia are kg m². An object’s rotational inertia depends on the axis of rotation.

Next class: Rotational Dynamics…

• θ
• ω
• α

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• Torque: τ
• Rotational Inertia: I

• x
• v_x
• a_x

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• Force: F_x
• Mass: m

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Linear

Rotational Analogy

Newton’s Second Law:

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Rotational Inertia

Consider a body made of N particles, each of mass m_i, where i = 1 to N. Each particle is located a distance r_i from the axis of rotation. For this body made of a countable number of particles, the rotational inertia is:

The units of rotational inertia are kg m². An object’s rotational inertia depends on the axis of rotation.

Four small metal spheres, each with mass 0.2 kg, are arranged in a square 0.40 m on a side and connected by extremely light rods.

Find the rotational inertia about an axis through the centre of the square, perpendicular to its plane.

Four small metal spheres, each with mass 0.2 kg, are arranged in a square 0.40 m on a side and connected by extremely light rods.

Find the rotational inertia about an axis through the centre of the square, parallel to its plane.

Figure 10.11: Rotational Inertias of Simple Objects

Figure 10.11: Rotational Inertias of Simple Objects

Figure 10.11: Rotational Inertias of Simple Objects

Figure 10.11: Rotational Inertias of Simple Objects

Figure 10.11: Rotational Inertias of Simple Objects

Figure 10.11: Rotational Inertias of Simple Objects

From today’s Preclass Survey

Rotational Dynamics

• From Chapter 9, we know how to keep things in Static Equilibrium. (zero net torque, zero net force)
• But what does it take to get something rotating?
• And what does it take to stop something from rotating?

Observational Experiment #1

Observational Experiment #2

Observational Experiment #3

Observational Experiment #4

Rotational form of Newton’s Second Law

• One or more objects exert forces on a rigid body with rotational inertia I that can rotate about some axis.
• The net torque due to these forces about that axis causes the object to have a rotational acceleration α:

Analogy chart

• θ
• ω
• α

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• Torque: τ
• Rotational Inertia: I

• x
• v_x
• a_x

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• Force: F_x
• Mass: m

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Linear

Rotational Analogy

Newton’s Second Law:

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Massive Pulleys

• A real pulley has some rotational inertia, I_pulley.
• This means that it will require a net torque in order to cause it to have rotational acceleration.
• For an accelerating system, the tension must change as it goes over the pulley.

I_pulley

The Massive Pulley Constraints

When an object is attached to a pulley, the distance the object travels is equal to the change in angular position of the pulley times the radius of the pulley.

• Distance m₂ and m₁ move is: s = θR, where θ is the angle the pulley turns through.
• Speeds of m₂ and m₁ are related to the angular speed of the pulley by: v = ωR
• Accelerations of m₂ and m₁ are related to the angular acceleration of the pulley by:

a = αR

I_pulley

Angular Momentum

• Angular momentum � = moment of inertia × angular velocity

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• This is analogous to

Linear momentum = mass × velocity

“The Right Hand Rule”

• The magnitude of the angular velocity vector is ω.
• The angular velocity vector points along the axis of rotation in the direction given by the right-hand rule as illustrated.

• Angular momentum is in the same direction as angular velocity.

Conservation of Angular Momentum

An isolated system that experiences no net torque has

L_f = L_i or I_f ω_f = I_i ω_i

Demonstration�

• Which has greater moment of inertia, I around a central vertical axis of rotation?
• Chair + Harlow + two 5 kg masses, arms outstretched
• Chair + Harlow + two 5 kg masses held near chest
• Assume external torque = zero, so L = I ω is constant.
• Harlow begins rotating at some particular value of ω is with his arms outstretched.
• He then brings the masses in to his chest.
• How does this affect the rotation speed ω?

Demonstration

• Harlow begins rotating at some particular value of ω is with his arms outstretched. I is large, ω is small, L = I ω is some value.
• He then brings the masses in to his chest. I decreases.
• But there is no external torque, so L = I ω value must stay the same.
• So ω is increases.

Top Hat Question 6

• Harlow is not rotating. He is holding a bicycle wheel which is rotating counterclockwise as viewed from above. What is the direction of the rotational momentum of the chair + Harlow + bicycle wheel system?
• Up
• Down
• Assume no external torque.
• If Harlow flips the wheel upside down, it is now rotating clockwise as viewed from above.
• The rotational momentum of the wheel is now down.
• What happens to Harlow+chair?

Demonstration Recap

• Harlow is not rotating. He is holding a bicycle wheel which is rotating counterclockwise as viewed from above.
• Rotational momentum of the system is up.
• Harlow flips the wheel upside down, so its rotational momentum is now down.
• No external torques, so the rotational momentum of the system must still be up.
• Harlow + chair rotate counterclockwise.

No chair rotation

Example:

Mass m₂ = 2.5 kg is on a frictionless table. It is attached to a cord that wraps around a massive pulley of radius R = 0.20 m, and rotational inertia I = 0.050 kg m². The same cord is attached to a hanging mass m₁ = 0.75 kg. After the system is released, what is the acceleration of m₁ as it falls?

The Cross Product Direction

Magnitude:

Cross Product Examples

Cross Product Examples

The magnetic force provides a centripetal force which causes charged particles to move in circles, or helixes.

This is what causes the Northern Lights!

Cross Product Examples

• We have been describing torque as “counter-clockwise” or “clockwise”, giving it a +/- sign to specify which.
• In general, the 3-dimensional definition of torque is that it is a vector, as defined by the cross product:

Rotational Impulse

Where:

Angular momentum of an isolated system is constant

• If the net torque that external objects exert on a turning object is zero, or if the torques add to zero, then the angular momentum L of the turning object remains constant:

L_f = L_i or I_f ω_f = I_i ω_i

A 20-cm radius, 2.0 kg solid disk is rotating at 200 rpm. A 20-cm-radius, 1.0 kg circular loop is dropped straight down onto the rotating disk. Friction causes the loop to accelerate until it is “riding” on the disk. What is the final angular speed of the combined system?

Rotational Kinetic Energy

A rotating rigid body has kinetic energy because all atoms in the object are in motion. The kinetic energy due to rotation is called rotational kinetic energy.

From today’s Preclass Survey

Flywheels for storing and providing energy

• In a car with a flywheel, instead of rubbing a brake pad against the wheel and slowing it down, the braking system converts the car's translational kinetic energy into the rotational kinetic energy of the flywheel.
• As the car's translational speed decreases, the flywheel's rotational speed increases. This rotational kinetic energy could then be used later to help the car start moving again.

https://www.greencarreports.com/news/1042570_porsche-911-hybrid-test-car-uses-flywheel-to-store-energy

Complete Linear / Rotational Analogy Chart

• θ, ω, α
• Torque: τ
• Moment of Inertia: I

• , ,
• Force:
• Mass: m

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Linear

Rotational Analogy

• Newton’s 2^nd law:
• Kinetic energy:
• Momentum:

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Summary of some Different Types of Energy:

• Kinetic Energy due to bulk motion of centre of mass: KE = ½ mv² (Sometimes called Translational Kinetic Energy KE_tran)
• Gravitational Potential Energy (System = object + Earth): PE_g = mgy
• Spring Potential Energy (system = object + spring): PE_s = ½ kx²
• Rotational Kinetic Energy: KE_rot = ½ Iω²
• Internal Thermal Energy: ΔE_int (If the system includes two surfaces rubbing against each other, then ΔE_int = | f_kd | )
• A system can possess any or all of the above.
• One way of transferring energy in or out of a system is work:
• Work done by a constant non-conservative force: W_nc = Fdcosθ

A 0.50 kg basketball rolls along the ground at 1.0 m/s. What is its total kinetic energy (translational plus rotational)? [Note that the rotational inertia of a hollow sphere is I = 2/3 MR².]

Solid Disk Vs Hoop.

• Demonstration: Solid Disk Wins.
• Instead of energy, let’s work this out with acceleration.

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Released from rest:

Solid disk rolls without slipping with greater acceleration than hoop!

1. What is the acceleration of a sliding object down a ramp inclined at angle θ? [assume no friction]

2. What is the acceleration of a round object rolling down a ramp inclined at angle θ? [assume rolling without skidding]

1. What is the acceleration of a Frictionless Ice Cube sliding down a ramp inclined at angle θ?

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• What is the acceleration of a Solid Disk rolling down a ramp inclined at angle θ? [assume rolling without skidding]

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• What is the acceleration of a Hoop rolling down a ramp inclined at angle θ? [assume rolling without skidding]

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• What is the acceleration of a Glass Marble rolling down a ramp inclined at angle θ? [assume rolling without skidding]

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HANDY RESULT:

Acceleration of a General rolling object down incline.

Gyroscopes

• Consider a solid disk on a long horizontal axle.
• It is supported by two wires on opposite sides of the axle, 2 m apart, with the disk in the middle.
• If the wire on the left breaks, the disk will pivot around the point where the unbroken wire on the right.
• Gravity pulls with a force F = mg downward on the centre of mass of the disk.
• R is the vector from the pivot to the center of mass.
• What is the direction of the gravitational torque on the disk?

Gyroscopes

Rotational impulse equals the change in angular momentum:

Gyroscopes

• Example
• A solid disk has a radius of R = 0.5 m, a mass of M = 1 kg, and is supported by a 2 m horizontal axle.
• The support at the left is removed.
• What is the gravitational torque around the right end of the axle?
• What will be the final angular momentum 0.1 seconds after it is released?
• Which way will the axle move?

Gyroscopes

• Example
• A solid disk has a radius of R = 0.5 m, a mass of M = 1 kg, and is supported by a 2 m horizontal axle. It is spinning at ω = 100 rad/s.
• The support at the left is removed.
• What is the initial angular momentum?
• If you add 0.98 kg m²/s of angular momentum out of the page, which way will the axle move?