Rotation and Circular

Motion

Linear vs. AngularLinear – in a straight line Angular – in a circle To connect the two…

x, y, d

Displacement

In meters

“how far…”

ΘIn radians (rad)

But can be rev, cycles, degreesAngular displacement

“how many revolutions…”1 rev = 2π rad = 360 º

𝑠 = 𝑟𝜃

s, v

In m/s

“How fast…”

ω

𝜔 =𝜃

𝑡In rad/s

Angular velocity – how fast in

a circle

𝑣 = 𝑟𝜔v is tangent to the circle

a

In m/s2

acceleration

α

𝛼 =∆𝜔

∆𝑡In rad/s2

Angular acceleration

𝑎 = 𝑟𝛼a is centripetal – “center-

seeking!)

When objects undergo constant

acceleration, relationships are the same:

Linear Rotational

𝑣𝑓 = 𝑣𝑖 + 𝑎𝑡

𝑣𝑓2 = 𝑣𝑖

2 + 2𝑎𝑥

𝑥 =1

2𝑣𝑖 + 𝑣𝑓 𝑡

𝑥 = 𝑣𝑖𝑡 +1

2𝑎𝑡2

Example1: A ball tied to a 0.65 m long

string turns twice in 4.2 seconds.

A. Calculate the angular velocity of the ball.

Calculate the tangential velocity of the ball.

A wheel turning at 2 rad/s accelerates at 3 rad/s2. How fast is it spinning after 5

seconds?

A wheel turning at 24 rad/s stops in 3.0 cycles. How long did it take to stop?

Centripetal Force (Fc) – the force that

keeps an object moving in a circle.

Centripetal Force causes centripetal acceleration! Both are center-

seeking! 𝐹𝑐 = 𝑚𝑎𝑐 In the swing example – the people would FLY out of the circle if the STRING

didn’t keep them there. The Centripetal Force is the TENSION in the string!

Centripetal Force is not a NEW force – it is whatever keeps the object

moving in a circle! (examples: Normal Force, Tension, Gravity…)

𝑎𝑐 =𝑣2

𝑟= 𝑟𝜔2

Draw the Free-Body Diagram / ID the

Centripetal Force…

Lets use this to find an unknown mass…

1. The 100 g stopper swings 50 times in one minute.

The cord length is 0.75 meters.

2. Calculate the angular velocity.

3. Draw the FBD for the stopper.

4. Write the Newton’s 2nd Law statement for the

stopper.

5. Use this to solve for the hanging mass!

Example problem: A 2200 kg car rounds a

curve that has a radius of 92.0 meters and a

coefficient of friction of 0.670.

A. Draw the FBD for the car

B. Calculate the Normal force on the car.

C. What is the max speed of the car in order to maintain control.

D. Identify and solve for the Centripetal Force on the car.

Torque – a force applied at a distance

can result in rotation

τ – (tau) the symbol we use for Torque

SI units are 𝑁 ∙ 𝑚

Vector quantity – clockwise positive

F – force in Newtons

l – lever arm (meters)

Measured from the fulcrum (pivot point)

Force and lever arm must be perpendicular!!!!!

𝜏 = 𝐹𝑙

Newton’s 2nd Law and rotation

𝐹 = 𝑚𝑎We can have net forces in the x, y, and centripetal

directions.

Centripetal forces happen when I’m going the SAME SPEED in a circle!

𝜏 = 𝐼𝛼

Torque makes objects rotate with angular acceleration.

Things go slower or faster in a circle.

Calculate the torque on the 0.42 m wrench with

a 17 N force that is 30.0 degrees from the

horizontal.

m1 = 35 kg r1 = 1.0 m

m2 = 22 kg r2 = ?

Newton’s Law of Universal Gravitation

Every object in the universe attracts every other

object with a force, as such:

G – universal gravitation constant

6.673 × 10−11𝑁 ∙ 𝑚2/𝑘𝑔2

m – masses in kg

r – distance between the centers of the masses

Used mostly for planets and atomic particles,

that orbit!

𝐹𝑔 = 𝐺𝑚1𝑚2

𝑟2

Mini lab!!! Calculate the attraction

between you and your desk neighbor!

1. Convert each of your weights to masses! 1 kg = 2.20 lb

2. Use a meterstick to measure the distance from your

waist to your neighbor’s waist.

3. Use Newton’s equation to calculate the gravitational

force between the two of you!

4. Now compare – you exert 1 N of force when you ring a

doorbell.

Example problems: Use Newton’s Law of

Gravitation to relate the following forces.

Keep all answers in terms of “F”

1. (Baseline) Two masses (m) that are r apart with a force of F.

2. One mass doubles.

3. Both masses are halved.

4. The distance is doubled.

5. Distance is one-quarter(ed), quadruple both masses.

6. Triple one mass, halve the other, halve the distance.