# AP Physics 1 : Angular Velocity and Acceleration

## Example Questions

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### Example Question #1 : Circular And Rotational Motion

A horizontally mounted wheel of radius  is initially at rest, and then begins to accelerate constantly until it has reached an angular velocity  after 5 complete revolutions. What was the angular acceleration of the wheel?

Explanation:

You may recall the kinematic equation that relates final velocity, initial velocity, acceleration, and distance, respectively:

Well, for rotational motion (such as in this problem), there is a similar equation, except it relates final angular velocity, intial angular velocity, angular acceleration, and angular distance, respectively:

The wheel starts at rest, so the initial angular velocity, , is zero. The total number of revolutions of the wheel is given to be 5 revolutions. Each revolution is equivalent to an angular distance of  radians. So, we can convert the total revolutions to an angular distance to get:

The final angular velocity was given as  in the text of the question. So, we should use the above equation to solve for the angular acceleration, .

### Example Question #2 : Angular Velocity And Acceleration

An object moves at a constant speed of  in a circular path of radius of 1.5 m. What is the angular acceleration of the object?

Explanation:

For a rotating object, or an object moving in a circular path, the relationship between angular acceleration and linear acceleration is

Linear acceleration is given by , angular acceleration is , and the radius of the circular path is .

For circular/centripetal motion, the linear acceleration is related to the object's linear velocity by

We know the linear velocity is , and the radius is 1.5 m, so we can find the linear acceleration...

Now that we have the linear acceleration, we can use this in the equation at the top to find the angular acceleration...

### Example Question #3 : Angular Velocity And Acceleration

If it takes a bike wheel 3 seconds to complete one revolution, what is the wheel's angular velocity?

Explanation:

The definition of angular velocity is

By identifying the given information to be  and , we can plug this into the equation to calculate the angular velocity:

### Example Question #4 : Angular Velocity And Acceleration

What is the angular velocity of the second hand of a clock?

Explanation:

The angular velocity of the second hand of a clock can be found by dividing the number of radians the second hand will travel over a known period of time. Thankfully for a clock, we know that the second hand will make one revolution, i.e. covering  in one minute, or 60s. The formula for angular velocity is:

So the angular velocity,  is , which simplifies to our answer,

### Example Question #5 : Angular Velocity And Acceleration

What is the difference in the angular velocity of the second hand of radius 1cm on a wristwatch, compared to the second hand of radius 5m on a large clock tower?

The clocktower second hand has an angular velocity that is 5 times faster than that of the wristwatch

The clocktower second hand has an angular velocity that is 500 times slower than that of the wristwatch

The clocktower second hand has an angular velocity that is 500 times faster than that of the wristwatch

The clocktower second hand has an angular velocity that is 20 times faster than that of the wristwatch

No difference

No difference

Explanation:

The angular velocity should not change based on the radius of the second hand. No matter what size the second hand, it will still cover one revolution every minute or 60s. The linear velocity will be greater and the angular momentum will also be greater for the clocktower, but its angular velocity will be the same. This can be seen by looking at the equation for angular velocity:

### Example Question #1 : Angular Velocity And Acceleration

A ferris wheel has a trip length of 3min, that is it takes three minutes for it to make one complete revolution. What is the angular velocity of the ferris wheel if it only takes passengers around one time, in ?

Explanation:

Angular velocity, in , is given by the length traveled divided by the time taken to travel the length:

We are told that the amount of time taken to make one revolution is 3min. One revolution is equal to , and 3 minutes is equal to 180 seconds. Divide the radian value by the seconds value to get the angular velocity.

### Example Question #2 : Circular And Rotational Motion

A wheel makes one full revolution every  seconds and has a radius of . Determine its angular velocity

Explanation:

For this question, the angular velocity  can be given by the equation:

, where  is the angle made and  is the time taken to make this angle.

In this problem, the wheel makes one full revolution() in  seconds.

Therefore:

### Example Question #8 : Angular Velocity And Acceleration

A CD rotates at a rate of  in the positive counter clockwise direction. After pressing play, the disk is speeding up at a rate of  . What is the angular velocity of the CD in  after 4 seconds?

Explanation:

Given initial angular velocity, angular acceleration, and time we can easily solve for final angular velocity with:

### Example Question #9 : Angular Velocity And Acceleration

If a ferris wheel has height of 100m, find the angular velocity in rotations per minute if the riders in the carts are going .

None of these

Explanation:

If the ferris wheel has height then it must have radius .

The circumference of the ferris wheel, or the distance of one rotation, is then:

Convert the given velocity into meters per minute, or :

Find rotations per minute:

### Example Question #10 : Angular Velocity And Acceleration

A person of mass is riding a ferris wheel of radius . The wheel is spinning at a constant angular velocity of . Determine the linear velocity of the rider.