# Precalculus : Angular and Linear Velocity

## Example Questions

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### Example Question #1 : Solve Angular Velocity Problems

If a ball is travelling in a circle of diameter  with velocity , find the angular velocity of the ball.

Explanation:

Using the equation,

where

=angular velocity, =linear velocity, and =radius of the circle.

In this case the radius is 5 (half of the diameter) and linear velocity is 20 m/s.

.

### Example Question #1 : Solve Angular Velocity Problems

Suppose a car tire rotates  times a second. The tire has a diameter of  inches. Find the angular velocity in radians per second.

Explanation:

Write the formula for angular velocity.

The frequency of the tire is 8 revolutions per second. The radius is not used.

Substitute the frequency and solve.

### Example Question #1 : Solve Angular Velocity Problems

What is the angular velocity of a spinning top if it travels  radians in a third of a second?

Explanation:

Write the formula for average velocity.

The units of omega is radians per second.

Substitute the givens and solve for omega.

### Example Question #1 : Solve Angular Velocity Problems

A  diamter tire on a car makes  revolutions per second. Find the angular speed of the car.

Explanation:

Recall that  .

Since the tire revolves 9.3 times/second it would seem that the tire would rotate

or .

We use  to indicate that the tire is rolling 360 degrees or  radians each revolution (as it should).

Thus,

Note that radians is JUST a different way of writing degrees. The higher numbers in the answers above are all measures around the actual linear speed of the tire, not the angular speed.

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

A car wheel of radius 20 inches rotates at 8 revolutions per second on the highway. What is the angular speed of the tire?

None of these.

Explanation:

Angular speed is the same as linear speed, but instead of distance per unit time we use degrees or radians. Any object traveling has both linear and angular speed (though objects only have angular speed when they are rotating).

Since our tire completes 8 revolutions per second we multiply by  since a full rotation (360°) equals .

### Example Question #1 : Solve Linear Velocity Problems

A plane is traveling Northeast. If the eastern component of it's velocity is , how fast is the plane traveling?

Explanation:

Since the plane is traveling NE, we know that it is traveling at an angle of 45 degrees north of east. Therefore, we can represent this in a diagram:

We have all of the information we need to solve for the velocity of the plane. We have an angle and a side adjacent to that angle. Therefore, we can use the cosine function to solve for the velocity.

Rearranging for v, we get:

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

A car is traveling NW at a speed of  mph. What are the magnitudes of each of its North and West components?

North:  mph

West:  mph

North:  mph

West:  mph

North:  mph

West:  mph

North:  mph

West:  mph

North:  mph

West:  mph

North:  mph

West:  mph

Explanation:

If the car is traveling NW, it means that the angle  is  degrees above the x-axis. Using this angle, and the fact that the magnitude of the velocity is  mph, we can find the x and y components, or the west and north components using trigonometry.

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

An object is moving in a circular path. How will its linear velocity change if the diameter of the circular path was decreased by one half?

Explanation:

Write the formula for the linear velocity.

The diameter is twice the radius.

If the diameter was decreased by one half, the radius will also decrease by one half.

Therefore:

The linear velocity will decrease by one half.

### Example Question #61 : Trigonometric Functions

Find the linear velocity in meters per second of an object if it took  hours to travel a distance of  kilometers. Round to the nearest integer.

Explanation:

Write the linear velocity formula.

Since the answer is required in meters per second, two conversions will need to be made.

Convert 30 kilometers to meters.

Convert 2 hours to seconds.

Apply the formula.

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

The second hand of a clock is  long. Find the linear speed of the end of this second hand.

Explanation:

Linear velocity is defined as distance over a period of time. For instance if a person ran 1 mile or approximately 1600 meters in 7 minutes, the they would have covered about 230 meters per minute. Let's assume this person was running around a track. We could also measure their speed from a central angle and represent their speed as the amount of degrees (or radians) they ran around per unit time as well. This is considered angular speed. A perfect example of both are the hands on a clock. There is a relationship between arc length which we designate s, the radius r, and the angle  (theta). The relationship is . So the length of the arc (s) is equal to the radius of the circle the arc is on and what section of the pie it covers ( this is akin to how many degrees our track runner ran "through" or around) . Thus, if we wanted the linear speed around a circle we could say  or . Key measurements that you will need to know are how many degrees in a circle of which there are 360 or . Before you follow the step by step solution below, go back and see if you can use this new information to arrive at the correct answer.