 ## Learning Objectives

### Learning Objectives

By the end of this section, you will be able to do the following:

• Describe the angle of rotation and relate it to its linear counterpart
• Describe angular velocity and relate it to its linear counterpart
• Solve problems involving angle of rotation and angular velocity
 angle of rotation angular velocity arc length circular motion radius of curvature rotational motion spin tangential velocity

## Angle of Rotation

### Angle of Rotation

What exactly do we mean by circular motion or rotation? Rotational motion is the circular motion of an object about an axis of rotation. We will discuss specifically circular motion and spin. Circular motion is when an object moves in a circular path. Examples of circular motion include a race car speeding around a circular curve, a toy attached to a string swinging in a circle around your head, or the circular loop-the-loop on a roller coaster. Spin is rotation about an axis that goes through the center of mass of the object, such as Earth rotating on its axis, a wheel turning on its axle, the spin of a tornado on its path of destruction, or a figure skater spinning during a performance at the Olympics. Sometimes, objects will be spinning while in circular motion, like the Earth spinning on its axis while revolving around the Sun, but we will focus on these two motions separately.

When solving problems involving rotational motion, we use variables that are similar to linear variables (distance, velocity, acceleration, and force) but take into account the curvature or rotation of the motion. Here, we define the angle of rotation, which is the angular equivalence of distance; and angular velocity, which is the angular equivalence of linear velocity.

When objects rotate about some axis—for example, when the CD in Figure 6.2 rotates about its center—each point in the object follows a circular path.

Figure 6.2 All points on a CD travel in circular paths. The pits (dots) along a line from the center to the edge all move through the same angle $ΔθΔθ$ in time $ΔtΔt$.

The arc length, , is the distance traveled along a circular path. The radius of curvature, r, is the radius of the circular path. Both are shown in Figure 6.3.

Figure 6.3 The radius (r) of a circle is rotated through an angle $ΔθΔθ$. The arc length, $ΔsΔs$, is the distance covered along the circumference.

Consider a line from the center of the CD to its edge. In a given time, each pit (used to record information) on this line moves through the same angle. The angle of rotation is the amount of rotation and is the angular analog of distance. The angle of rotation $ΔθΔθ$ is the arc length divided by the radius of curvature.

$Δθ=ΔsrΔθ=Δsr$

The angle of rotation is often measured by using a unit called the radian. (Radians are actually dimensionless, because a radian is defined as the ratio of two distances, radius and arc length.) A revolution is one complete rotation, where every point on the circle returns to its original position. One revolution covers $2π2π$ radians (or 360 degrees), and therefore has an angle of rotation of $2π2π$ radians, and an arc length that is the same as the circumference of the circle. We can convert between radians, revolutions, and degrees using the relationship

1 revolution = $2π2π$rad = 360°. See Table 6.1 for the conversion of degrees to radians for some common angles.

6.1
Degree Measures Radian Measures
$30∘30∘$ $π6π6$
$60∘60∘$ $π3π3$
$90∘90∘$ $π2π2$
$120∘120∘$ $2π32π3$
$135∘135∘$ $3π43π4$
$180∘180∘$ $ππ$
Table 6.1 Commonly Used Angles in Terms of Degrees and Radians

## Angular Velocity

### Angular Velocity

How fast is an object rotating? We can answer this question by using the concept of angular velocity. Consider first the angular speed $(ω)(ω)$ is the rate at which the angle of rotation changes. In equation form, the angular speed is

6.2$ω=ΔθΔt,ω=ΔθΔt,$

which means that an angular rotation $(Δθ)(Δθ)$ occurs in a time, $ΔtΔt$. If an object rotates through a greater angle of rotation in a given time, it has a greater angular speed. The units for angular speed are radians per second (rad/s).

Now let’s consider the direction of the angular speed, which means we now must call it the angular velocity. The direction of the angular velocity is along the axis of rotation. For an object rotating clockwise, the angular velocity points away from you along the axis of rotation. For an object rotating counterclockwise, the angular velocity points toward you along the axis of rotation.

Angular velocity (ω) is the angular version of linear velocity v. Tangential velocity is the instantaneous linear velocity of an object in rotational motion. To get the precise relationship between angular velocity and tangential velocity, consider again a pit on the rotating CD. This pit moves through an arc length $(Δs)(Δs)$ in a short time $(Δt)(Δt)$ so its tangential speed is

6.3 $v=ΔsΔt.v=ΔsΔt.$

From the definition of the angle of rotation, $Δθ=ΔsrΔθ=Δsr$, we see that $Δs=rΔθΔs=rΔθ$. Substituting this into the expression for v gives

$v=rΔθΔt=rω.v=rΔθΔt=rω.$

The equation $v=rωv=rω$ says that the tangential speed v is proportional to the distance r from the center of rotation. Consequently, tangential speed is greater for a point on the outer edge of the CD (with larger r) than for a point closer to the center of the CD (with smaller r). This makes sense because a point farther out from the center has to cover a longer arc length in the same amount of time as a point closer to the center. Note that both points will still have the same angular speed, regardless of their distance from the center of rotation. See Figure 6.4.

Figure 6.4 Points 1 and 2 rotate through the same angle ($ΔθΔθ$), but point 2 moves through a greater arc length ($Δs2Δs2$) because it is farther from the center of rotation.

Now, consider another example: the tire of a moving car (see Figure 6.5). The faster the tire spins, the faster the car moves—large $ωω$ means large v because $v=rωv=rω$. Similarly, a larger-radius tire rotating at the same angular velocity, $ωω$, will produce a greater linear (tangential) velocity, v, for the car. This is because a larger radius means a longer arc length must contact the road, so the car must move farther in the same amount of time.

Figure 6.5 A car moving at a velocity, v, to the right has a tire rotating with angular velocity $ωω$. The speed of the tread of the tire relative to the axle is v, the same as if the car were jacked up and the wheels spinning without touching the road. Directly below the axle, where the tire touches the road, the tire tread moves backward with respect to the axle with tangential velocity $v=rωv=rω$, where r is the tire radius. Because the road is stationary with respect to this point of the tire, the car must move forward at the linear velocity v. A larger angular velocity for the tire means a greater linear velocity for the car.

However, there are cases where linear velocity and tangential velocity are not equivalent, such as a car spinning its tires on ice. In this case, the linear velocity will be less than the tangential velocity. Due to the lack of friction under the tires of a car on ice, the arc length through which the tire treads move is greater than the linear distance through which the car moves. It’s similar to running on a treadmill or pedaling a stationary bike; you are literally going nowhere fast.

### Tips For Success

Angular velocity ω and tangential velocity v are vectors, so we must include magnitude and direction. The direction of the angular velocity is along the axis of rotation, and points away from you for an object rotating clockwise, and toward you for an object rotating counterclockwise. In mathematics this is described by the right-hand rule. Tangential velocity is usually described as up, down, left, right, north, south, east, or west, as shown in Figure 6.6.

Figure 6.6 As the fly on the edge of an old-fashioned vinyl record moves in a circle, its instantaneous velocity is always at a tangent to the circle. The direction of the angular velocity is into the page this case.

### Watch Physics

#### Relationship between Angular Velocity and Speed

This video reviews the definition and units of angular velocity and relates it to linear speed. It also shows how to convert between revolutions and radians.

Grasp Check
For an object traveling in a circular path at a constant speed, would the linear speed of the object change if the radius of the path increases?
1. Yes, because tangential speed is independent of the radius.
2. Yes, because tangential speed depends on the radius.
3. No, because tangential speed is independent of the radius.
4. No, because tangential speed depends on the radius.

## Solving Problems Involving Angle of Rotation and Angular Velocity

### Snap Lab

#### Measuring Angular Speed

In this activity, you will create and measure uniform circular motion and then contrast it with circular motions with different radii.

Materials
• One string (1 m long)
• One object (two-hole rubber stopper) to tie to the end
• One timer
Procedure
1. Tie an object to the end of a string.
2. Swing the object around in a horizontal circle above your head (swing from your wrist). It is important that the circle be horizontal!
3. Maintain the object at uniform speed as it swings.
4. Measure the angular speed of the object in this manner. Measure the time it takes in seconds for the object to travel 10 revolutions. Divide that time by 10 to get the angular speed in revolutions per second, which you can convert to radians per second.
5. What is the approximate linear speed of the object?
6. Move your hand up the string so that the length of the string is 90 cm. Repeat steps 2–5.
7. Move your hand up the string so that its length is 80 cm. Repeat steps 2–5.
8. Move your hand up the string so that its length is 70 cm. Repeat steps 2–5.
9. Move your hand up the string so that its length is 60 cm. Repeat steps 2–5
10. Move your hand up the string so that its length is 50 cm. Repeat steps 2–5
11. Make graphs of angular speed vs. radius (i.e. string length) and linear speed vs. radius. Describe what each graph looks like.
Grasp Check
If you swing an object slowly, it may rotate at less than one revolution per second. What would be the revolutions per second for an object that makes one revolution in five seconds? What would be its angular speed in radians per second?
1. The object would spin at $15rev/s$. The angular speed of the object would be $2π5rad/s$.
2. The object would spin at $15rev/s$. The angular speed of the object would be $π5rad/s$.
3. The object would spin at $5rev/s$. The angular speed of the object would be $10πrad/s$.
4. The object would spin at $5rev/s$. The angular speed of the object would be $5πrad/s$.

Now that we have an understanding of the concepts of angle of rotation and angular velocity, we’ll apply them to the real-world situations of a clock tower and a spinning tire.

### Worked Example

#### Angle of rotation at a Clock Tower

The clock on a clock tower has a radius of 1.0 m. (a) What angle of rotation does the hour hand of the clock travel through when it moves from 12 p.m. to 3 p.m.? (b) What’s the arc length along the outermost edge of the clock between the hour hand at these two times?

### Strategy

We can figure out the angle of rotation by multiplying a full revolution ($2π2π$ radians) by the fraction of the 12 hours covered by the hour hand in going from 12 to 3. Once we have the angle of rotation, we can solve for the arc length by rearranging the equation $Δθ=ΔsrΔθ=Δsr$ since the radius is given.

Solution to (a)

In going from 12 to 3, the hour hand covers 1/4 of the 12 hours needed to make a complete revolution. Therefore, the angle between the hour hand at 12 and at 3 is $14×2π rad=π214×2π rad=π2$ (i.e., 90 degrees).

Solution to (b)

Rearranging the equation

6.4$Δθ=Δsr,Δθ=Δsr,$

we get

6.5$Δs=rΔθ.Δs=rΔθ.$

Inserting the known values gives an arc length of

6.6$Δs=(1.0 m)(π2 rad)=1.6 mΔs=(1.0 m)(π2 rad)=1.6 m$
Discussion

We were able to drop the radians from the final solution to part (b) because radians are actually dimensionless. This is because the radian is defined as the ratio of two distances (radius and arc length). Thus, the formula gives an answer in units of meters, as expected for an arc length.

### Worked Example

#### How Fast Does a Car Tire Spin?

Calculate the angular speed of a 0.300 m radius car tire when the car travels at 15.0 m/s (about 54 km/h). See this figure.

### Strategy

In this case, the speed of the tire tread with respect to the tire axle is the same as the speed of the car with respect to the road, so we have v = 15.0 m/s. The radius of the tire is r = 0.300 m. Since we know v and r, we can rearrange the equation $v=rωv=rω$, to get $ω=vrω=vr$ and find the angular speed.

Solution

To find the angular speed, we use the relationship: $ω=vrω=vr$.

Inserting the known quantities gives

6.7$ω=15.0 m/s0.300 m=50.0 rad/s.ω=15.0 m/s0.300 m=50.0 rad/s.$
Discussion

When we cancel units in the above calculation, we get 50.0/s (i.e., 50.0 per second, which is usually written as 50.0 s−1). But the angular speed must have units of rad/s. Because radians are dimensionless, we can insert them into the answer for the angular speed because we know that the motion is circular. Also note that, if an earth mover with much larger tires, say 1.20 m in radius, were moving at the same speed of 15.0 m/s, its tires would rotate more slowly. They would have an angular speed of

6.8$ω=15.0 m/s1.20 m=12.5 rad/sω=15.0 m/s1.20 m=12.5 rad/s$

## Practice Problems

### Practice Problems

What is the angle in degrees between the hour hand and the minute hand of a clock showing 9:00 a.m.?

1. 90°
2. 180°
3. 360°

What is the approximate value of the arc length between the hour hand and the minute hand of a clock showing 10:00 a.m. if the radius of the clock is 0.2 m?

1. 0.1 m
2. 0.2 m
3. 0.3 m
4. 0.6 m

## Check Your Understanding

### Check Your Understanding

Exercise 1
What is circular motion?
1. Circular motion is the motion of an object when it follows a linear path.
2. Circular motion is the motion of an object when it follows a zigzag path.
3. Circular motion is the motion of an object when it follows a circular path.
4. Circular motion is the movement of an object along the circumference of a circle or rotation along a circular path.
Exercise 2

What is meant by radius of curvature when describing rotational motion?

1. The radius of curvature is the radius of a circular path.
2. The radius of curvature is the diameter of a circular path.
3. The radius of curvature is the circumference of a circular path.
4. The radius of curvature is the area of a circular path.
Exercise 3
What is angular velocity?
1. Angular velocity is the rate of change of the diameter of the circular path.
2. Angular velocity is the rate of change of the angle subtended by the circular path.
3. Angular velocity is the rate of change of the area of the circular path.
4. Angular velocity is the rate of change of the radius of the circular path.
Exercise 4
What equation defines angular velocity, $ω$? Take that $r$ is the radius of curvature, $θ$ is the angle, and $t$ is time.
1. $ω=ΔθΔt$
2. $ω=ΔtΔθ$
3. $ω=ΔrΔt$
4. $ω=ΔtΔr$
Exercise 5
Identify three examples of an object in circular motion.
1. an artificial satellite orbiting the Earth, a race car moving in the circular race track, and a top spinning on its axis
2. an artificial satellite orbiting the Earth, a race car moving in the circular race track, and an electron moving in the circular orbit around the nucleus
3. Earth spinning on its own axis, a race car moving in the circular race track, and an electron moving in the circular orbit around the nucleus.
4. Earth spinning on its own axis, blades of a working ceiling fan, and a top spinning on its own axis.
Exercise 6
What is the relative orientation of the radius and tangential velocity vectors of an object in uniform circular motion?
1. Tangential velocity vector is always parallel to the radius of the circular path along which the object moves.
2. Tangential velocity vector is always perpendicular to the radius of the circular path along which the object moves.
3. Tangential velocity vector is always at an acute angle to the radius of the circular path along which the object moves.
4. Tangential velocity vector is always at an obtuse angle to the radius of the circular path along which the object moves.