Rotational Motion Notes

Spinning objects and objects that move in a circular path fall into a category of physics we call rotational motion. The beauty and elegance of rotational motion is that it shares sort of a 90% similarity with the aspects of linear motion that we have already studied.

Angular Displacement (theta)

An object’s position on a circular path, basically how far it has rotated. Must be measured in RADIANS.

Radian Measurement: A way to measure distances around a circle, radians are based on the pi () ratio and describe units of arc lengths in terms of the radius of the circle.

Angular Velocity  (omega)

The rate at which an object rotates. The rate of change of angular displacement. Units: rad/sec

Angular Acceleration (alpha)

The rate at which an object increases (or decreases) its rate of rotation. Units: rad/sec2

Using these relationships we can then say:

Relationships between linear and angular variables:

Think about driving your car down the street. The fast your tires spin, the faster your car does forward under normal conditions, right? There is actually a very simple relationship between these quantities and the radius of the circle:

Linear / Angular / Relationship
x /  / x = r
v /  / v = r
a /  / at = r

Note the subscript t on the acceleration. This is what we call “tangential acceleration” and is related to speeding up the rate of rotation. In a few days we will learn about “centripetal force” and “centripetal acceleration” which is very important to circular movement, but different than this acceleration. We will not be solving any problems using tangential acceleration.

Rotational Motion Problems:

  1. A child’s top rotates through 22 revolutions every minute. Determine the angular velocity in radians per second.
  1. A car’s engine idles at 1500 rev/min. This describes the number of times that the engine cycles thro0ugh each cylinder each minute. Each “cycle” is sort of like a revolution of the engine. Convert this cycle rate to rad/s.
  1. A car wheel rotates 256 rev/min. Assume the radius of the wheels is 0.5m.
  2. What is the angular speed of the wheel?
  3. How fast is the car going?
  4. What is the angular displacement of the wheel during 30 seconds of driving?
  5. How far has the car gone?
  1. Starting from rest a circular saw blade reaches a rotational speed of 600 rev/min in a time of 1.2 seconds. Determine the angular acceleration of the saw blade.
  1. A basketball player spins the ball on his finger. A fan notices that the ball makes 15 rev in 4 seconds. What is the angular velocity of the ball in

a)rev/s

b)rad/s

Graphical Relationships

The angular variables of motion share with each other the same relationships as the linear variables. Fill in the following definitions:

Angular velocity is the ______of the graph of angular displacement vs. time.

Angular ______is the slope of the graph of angular velocity vs. time.

In the space provided, sketch graphs of angular displacement and velocity that represent the motion described:

a)A car’s wheel while the car drives at a constant speed.

b)A merry go round while the children are pushing it to go faster

c)A merry go round where all the children are on and no one is pushing

How would the graphs of linear displacement and velocity vs. time compare to the ones that you drew for each of these scenarios? Explain.

Centripetal Acceleration & Force

A commonly mispronounced and misunderstood force, there is always a centripetal force associated with any type of rotational motion. A common way to experience this force is to consider what happens to your car and your body as you turn a sharp corner at a high rate of speed. We will discuss this shortly.

Definition: Centripetal Force

Not “centrifugal force” This is something entirely different, actually the opposite of the force we want to talk about and opens up a discussion that is far more complicated than any first-year physics student need worry about.

Centripetal forces are always caused by other forces. It is not a special type of force, it is just a special mathematical case where we can assign a unique mathematical form to forces that we already know about.

Some centripetal forces:

During the spin cycle of a washing machine the clothes are sun until they press against the wall of the washer drum, forcing water out of the clothes and out of the holes in the drum. Name the force that holds the clothes in the circle and keeps them moving in their circular path? What is the direction of that force?

The moon orbits the Earth in a nearly circular orbit. What force keeps the moon orbiting our planet? The same is true for the Earth orbiting the sun. What is the direction of that force?

A foolish student of physics decides it would be cool if he attached some padlocks to the end of a chain and whirls them around his head. What force keeps the padlocking moving in a circle? What is the direction of that force?

You car rounds a sharp corner at a high rate of speed. You are in the backseat. What happens to you? What keeps you in the car? What is the direction of that force?

Your car rounds a corner at high speed. What keeps your car moving around the turn? Before you answer this question, think about what happens when the road is wet, or if there is loose gravel on the corner. What is the direction of that force?

A physics student with a jeep (no doors) decides to do an experiment. He covers his seat with plastic and coats the plastic with butter. He does not wear his seatbelt. He drives up to a corner and turns sharply. What happens to him. Explain. Use the words “centripetal force”

The centripetal force always points ______.

How to calculate the centripetal force:

Fc = mac

1)Find your angular velocity.

2)Convert to radians per second

3)Convert to a linear velocity using the “relationship” equations

4)Solve for centripetal acceleration

5)Multiply by mass to get centripetal force

Most of your problems will start with a linear velocity given and you simply have to calculate the acceleration and the force from that (#’s 1 & 2), but some will be a little more complicated (#’s 3 & 4).

  1. A car around a corner at 12 m/s. The car stays about 10 m from a light post just inside the curb as it swings around the corner.

a)What is the radius of the arc swung out by the car?

b)What is the centripetal acceleration?

c)What is the centripetal force?

d)What force causes this centripetal force (what keeps the car in the turn)?

  1. The clothes in your washer press against the drum with a force of 30 N. If a wet t-shirt has a mass of 0.75 kg, and the radius of the washer drum is 0.5 m, what is the velocity of the clothes as they spin?

What is the angular velocity of the drum?

  1. It takes about 20,000 N of force to keep a 1000kg car on the road on a racetrack curve. Assuming the radius of the turn in 22m, what is the maximum velocity necessary to keep a car on the turn?
  1. At the NASA space center, astronauts are placed into a training module at the end of a large metal arm. The arm and module are rotated, producing extreme accelerations which prepare the astronaut’s body for the magnitudes of the forces it will experience during liftoff and re-entry. A human can usually withstand about 12 g’s (or 12 x 9.81 m/s^2 = about 118 m/s^2) of acceleration for a few moments before blacking out. The length of the arm is about 3m long with the astronaut on the end of the arm. What angular velocity of this mechanism would be necessary to produce this centripetal force with this apparatus?
  1. The rotating swings ride at Six Flags rotates 15 times per minute. Passengers are swung at the end of metal chains that place them about 12 m from the center of the ride. Assuming the force in the chains is all centripetal, determine

a)the centripetal acceleration of the passengers

b)the tension in a chain supporting a 55 kg adult

c)the tension in a chain supporting a 25 kg child