 # Angular Speed Questions

## Angular Speed

Speed is fundamentally a quantity that shows how fast or slow a body is moving. Similarly, angular speed is a speed that confirms how quickly a body rotates. In other words, it is defined as the change in the angle of a body per unit of time.

Therefore, to compute the speed of the rotational motion of a body, we need the angular speed of it. Angular speed formula calculates the distance covered by the body in terms of revolutions to the time taken.

Also, radian is quite an important thing here. Every time we calculate the angular speed, the angle we measure is in radians. Radians is a method of measuring angles where we describe the right angle as pi/2 radians. Therefore, one complete revolution will cover around 6.28 radians.

## Formula

We know the rate at which an object changes its angles is the angular speed, which we measure in radians in a given time. Angular speed has a magnitude only.

ω = θ /t

Here:

ω denotes to the angular speed in radians/sec

t denotes to the time, sec

It is important to note that while calculating angular speed and angular velocity the same formula is used. Though, the difference between these two is that angular velocity is a vector quantity and angular speed is scalar quantity.

### Important Questions on Angular Speed

1) If a ball is travelling in a circle of diameter 20m with velocity 40m/s, then calculate the angular velocity of the ball.

Using the equation,

ω=vr where

ω=angular velocity, v=linear velocity, and r=radius of the circle.

In this case the radius is 10 and linear velocity is 40 m/s.

ω=40/10=4.

2) What will be the angular velocity of a rotating top if it travels π radians in a third of a second?

Average velocity ω=θt

The units of omega are radians per second.

Substitute the givens and solve for omega.

ω=θ/t=π/ (1/3) = π∗3 =3π

3) Assume that a car tyre rotates 10 times a second. The tyre has a diameter of 10 inches. Find the angular velocity in radians per second.

Write the formula for angular velocity.

ω=2πf

The frequency of the tyre is 10 revolutions per second. The radius is not used.

Substitute the frequency and solve.

ω=2πf=2π (10) =20π

4) A car wheel of radius 30 inches rotates at 10 revolutions per second on the highway. Find the angular speed of the tyre.

Angular speed is similar to linear speed, but as a replacement for distance per unit time, we use degrees or radians.

Angular speed= θ/t

Since our tyre completes 10 revolutions per second, we multiply by 2π since a full rotation (360°) equals 2π.

5) A car tyre of diameter 20 makes 9.3 revolutions per second. Calculate the angular speed of the car.

Angular velocity=θ/t.

Since the tyre revolves 9.3 times/second

We use 2π to indicate that the tyre is rolling 360 degrees or 2π radians each revolution

Thus,

6) What is the similarity and difference between Angular Speed and Angular Velocity?

These two quantities have the following similarities and differences.

1. Similarity – Both are measured in radians per second.

2. Difference – Angular speed (v) is a scalar quantity and Angular velocity (ω) is a vector quantity.

7) What is the angle between the Acceleration Vector and Velocity Vector?

Depending on the following types of motion, the angle between the velocity and acceleration vector differs.

• If circular motion exists then, the angle is 90⁰.
• In linear motion, the angle may be 0⁰ or 180⁰ depending upon the increase or decrease in the speed.

8) A car tyre revolves at 180 rpm. Find its Angular Velocity in rad/sec.

1 revolution = 2 π radians

180 revolutions/minute = (180 x 6.28) rev/min

= (180 x 6.28)/60 = 18.84 rev/sec

9) Determine the angular velocity if 4.6 revolutions are completed in 4 seconds.

The angular displacement is 4.6 x 2 π or 9.2 π radians.

Since, ω = θ /t

= ω = 9.2 π/4

10) Determine the linear velocity of a point rotating at an angular velocity of 16 π radians per second at a distance of 8cm from the centre of the rotating object.

v = r ω

v = (8) (16 π) where, r = 8, ω = 16 π

= v = 402.1 cm per second.

### Practice Questions

1. What is the relationship between angular speed and linear speed?
2. Is angular speed a vector or scalar quantity?
3. Define angular speed and angular acceleration.
4. What will be the angular speed for 1 revolution?
5. An object is performing a uniform circular motion with an angular speed of π/12 of radian per second. At t = 0 the object starts at an angle θ =0. Find the angular displacement of the particle after 6s.

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