AP PHYSICS 1 ROTATIONAL MOTION/ANGULAR MOMENTUM

Translational Motion

Type of motion where an object starts at one point and moves to another.

Rotational Motion

Type of motion where an object "spins in place".

Angular Displacement

A measure of the amount of radians that an object rotates through.

Angular Velocity

A speed of rotation measured in radians/second.

Angular Acceleration

The rate of change of angular velocity in radians/second squared.

2 pi

The value you must multiply by to change revolutions to radians.

What is the slope of a graph of angular position vs time?

angular velocity

What is the slope of a graph of angular velocity vs time?

angular acceleration

Moment of Inertia

The rotational equivalent of mass, I

I (cylinder or disk)

I = 1/2 MR^2

I (solid sphere)

I = 2/5 MR^2

I (hollow sphere or ring)

I = 2/3 MR^2

I (Rotating rod at center)

I = 1/12 ML^2

I (Rotating rod at end)

I = 1/3 ML^2

Angular Momentum

The rotational equivalent of linear momentum. A product of moment of inertia and angular velocity, L

Law of Conservation of Angular Momentum

Angular Momentum is always conserved.

Rotation Fact

Every point on a rotating, rigid surface has the same angular velocity.

Frequency

The number of rpm, revolutions per minute. Can also be rotations or turns per minute or second.

If radius of a rotating object decreases, what happens?

Angular speed increases

Where is the greatest Moment of Inertia for a spinning object with changing radius?

Position where radius is the greatest

Where is the greatest angular Kinetic Energy for a spinning object with changing radius?

Position where angular speed is highest (smallest radius)

How do you calculate total kinetic energy for an object that is rotating about its axis and moving linearly?

Sum of : Translational KE (1/2 mv^2 ) +

Rotational KE (1/2 Iw^2)

What is angular acceleration?

Uniform circular motion: motion in a circular path at constant speed

How do we define angular velocity (omega)?

We defined angular velocity 𝜔 as the time rate of change of an angle theta.

What occurs with angular acceleration when change occurs?

There is an angular acceleration, in which 𝜔 changes. The faster the change occurs, the greater the angular acceleration.

-where delta omega is the change in angular velocity and delta t is the change in time.

The units of angular acceleration are rad/s2.

What happens if omega increases or decreases? What is the angular acceleration (alpha)?

Example: Suppose a teenager puts her bicycle on its back and starts the rear wheel spinning from rest to a final angular velocity of 250 rpm in 5.00 s. (a) Calculate the angular acceleration in rad/s2. (b) If she now slams on the brakes, causing an angular acceleration of -87.3rad/s2 , how long does it take the wheel to stop?

**Tangential acceleration and centripetal acceleration

What is linear or tangential acceleration?

Linear or tangential acceleration, 𝑎𝑡 refers to changes in the magnitude of velocity but not its direction. In circular motion centripetal acceleration, 𝑎𝑐 , refers to changes in the direction of the velocity but not its magnitude.

Tangential acceleration is directly related to?

Tangential acceleration alpha t is directly related to the angular acceleration alpha and is linked to an increase or decrease in the velocity, but not its direction.

What is the relationship between tangential acceleration a(t) and angular accretion (alpha)?

Example: A powerful motorcycle can accelerate from 0 to 30.0 m/s in 4.20 s. What is the angular acceleration of its 0.320 m radius wheels?

Rotational and Translational Quantities

**Kinematics of Rotational Motion

What is kinematics? What is kinematics of rotational motion?

Kinematics is the description of motion.

The kinematics of rotational motion describes the relationships among rotation angle, angular velocity, angular acceleration, and time.

Kinematics of Rotational Motion

Example: A deep-sea fisherman hooks a big fish that pulls line from his fishing reel, which is initially at rest. The reel is given an angular acceleration of 110 𝑟𝑎𝑑/𝑠2 for 2.00 𝑠 and line unwinds from the reel at a radius of 4.50 cm from its axis of rotation.

(a) What is the final angular velocity of the reel?

(b) At what speed is fishing line leaving the reel after 2.00 s elapses?

(c) How many revolutions does the reel make?

**Dynamics of Rotational Motion: Rotational Inertia

What is the relationship between force, mass, radius and angular acceleration?

To develop the precise relationship among force, mass, radius, and angular acceleration, consider what happens if we exert a force F on a point mass m that is at a distance r from a pivot point

What is the equation that shows this relationship?

This equation is the rotational analog of Newton's second law (F=ma ) Torque is analogous to force

Angular acceleration is analogous to translational acceleration mr^2 is analogous to mass (or inertia)

**Moment of Inertia and Torque

What is the equation for torque, moment of intertia, and angular acceleration?

The units of moment of intertia are kgm^2

(If you want to give something angular acceleration - you need torque).

Main idea of rotational inertia

Example: A father pushing a playground merry-go-round. He exerts a force of 250 𝑁 at the edge of the 50.0 𝑘𝑔 merry-go-round, which has a 1.50 𝑚 radius. Calculate the angular acceleration produced when no one is on the merry-go-round. Consider the merry-go-round itself to be a uniform disk.

Rotational Inertia

**Rotational Kinetic Energy: Work and Energy Revisited

net work (W) = net Fd cos(theta)

Work energy theorem for rotational motion only

Rotational kinetic energy KE(rot)

**Angular Momentum and Its Conservation

How is angular momentum defined?

We define angular momentum L as

L = Iw

As we would expect, an object that has a large moment of inertia I, such as Earth, has a very large angular momentum.

What happens to angular momentum when net torque is zero?

What is the equation for the law of conservation of angular momentum?

**Conservation of angular momentum

Example of Conservation of angular momentum

Example: Suppose an ice skater is spinning at 0.800 𝑟𝑒𝑣/ 𝑠 with her arms extended. She has a moment of inertia of 2.34𝑘𝑔 ⋅ 𝑚2 with her arms extended and of 0.363𝑘𝑔 ⋅ 𝑚2 withherarmsclosetoherbody.(a)Whatisherangularvelocityinrevolutionsper second after she pulls in her arms? (b) What is her rotational kinetic energy before and after she does this?

circumference of a circle

angular displacement

angle the object rotates through during some time interval (θ- theta)

equation for angular displacement (θ)

s=arc length

r= radius

units: radians

to convert degrees to radians and vice versa

multiply by π/180 or 180/π

angular speed

ratio of angular displacement to time interval (ω- omega)

equation for angular speed (ω)

units: rad/s

instantaneous angular speed

the limit of the average speed as the time interval approaches zero

avg. angular acceleration

the ratio of the change in the angular speed to the time it takes for the object to undergo the change (α- alpha)

equation for angular acceleration (α)

units: rad/s²

rotational motion kinematics equations

ω=ωi + αt

∆θ=ωit + 1/2αt²

ω²=ωi² + 2α∆θ

period

time it takes for object in circular motion to make 1 complete revolution, or cycle

-measured in seconds

frequency

number of cycles in 1 second

-measured in 1/sec or sec⁻¹, aka Hz (Hertz)

f=1/T

equation #1 for centripetal acceleration

equation #2 for centripetal acceleration

a= rω²

forces that cause centripetal acceleration

-level curves

-banked curves

-horizontal curves

-vertical curves

level curves vs. banked curves

-level: friction is force that produces centripetal acceleration

-banked: component of normal force adds to frictional force to allow higher speeds

center of gravity

geometric center of object

-must lie on the axis of symmetry

how to find center of gravity

sum of each mass times its distance then divide by sum of masses