Angle of Rotation and Angular Velocity, Circular and Rotational Motion | Physics

Learning objectives

Section Key Terms

angle of rotation	angular velocity	arc length	circular motion
radius of curvature	rotational motion	spin	tangential velocity

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_06_01_CD rotates about its center—each point in the object follows a circular path.

View original figure (external; image not reproduced in this text edition)

Original figure description: The diagram shows a picture of a CD with about an eighth of the shiny surface peeled away to display the inside of the CD with pits (dots) arranged in lines from the center. An angle theta is marked from one line of dots to another.

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

Δ θ

in time

Δ t

The arc length, Δs, 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_06_01_RotationAngle.

View original figure (external; image not reproduced in this text edition)

Original figure description: The diagram shows a circle with the center marked O, a radius r ending on the circumference at A, an arc change of s starting at A and ending at B. The angle for the arc is change of theta. A formula to the right of the circle says change of theta is equal to change of s divided by r.

The radius (r) of a circle is rotated through an angle

Δ θ

. The arc length,

Δ s

, 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 $Δ θ$ is the arc length divided by the radius of curvature.

Δ θ = \frac{Δ s}{r}

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 $2 π$ radians (or 360 degrees), and therefore has an angle of rotation of $2 π$ 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 = $2 π$ rad = 360°. See Table_06_01 for the conversion of degrees to radians for some common angles.

\begin{matrix} 2 π rad & = & 360 ° \\ 1 rad & = & \frac{360 °}{2 π} \approx 57.3 ° \end{matrix}

Commonly Used Angles in Terms of Degrees and Radians
Degree Measures	Radian Measures
$30^{\circ}$	$\frac{π}{6}$
$60^{\circ}$	$\frac{π}{3}$
$90^{\circ}$	$\frac{π}{2}$
$120^{\circ}$	$\frac{2 π}{3}$
$135^{\circ}$	$\frac{3 π}{4}$
$180^{\circ}$	$π$

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 $(ω)$ is the rate at which the angle of rotation changes. In equation form, the angular speed is

ω = \frac{Δ θ}{Δ t},

which means that an angular rotation $(Δ θ)$ occurs in a time, $Δ t$ . 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 $(Δ s)$ in a short time $(Δ t)$ so its tangential speed is

v = \frac{Δ s}{Δ t} .

From the definition of the angle of rotation, $Δ θ = \frac{Δ s}{r}$ , we see that $Δ s = r Δ θ$ . Substituting this into the expression for v gives

v = \frac{r Δ θ}{Δ t} = r ω .

The equation $v = r ω$ 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_06_01_ArcLength.

View original figure (external; image not reproduced in this text edition)

Original figure description: The picture shows a circle with a radius r2 going from the center, through a point 1 half-way on the radius and point 2 on the circumference. The distance from center to point 1 is labeled r1. A second radius line is drawn with an angle of change in theta. The arc from point 1 to the second radius is labeled change s1. The arc at point 2 is labeled change s2. There are two formulas to the right of the diagram: theta is equal to change of s1 over r1 and theta is equal to change of s2 over r2.

Points 1 and 2 rotate through the same angle (

Δ θ

), but point 2 moves through a greater arc length (

Δ s_{2}

) because it is farther from the center of rotation.

Now, consider another example: the tire of a moving car (see Figure_06_01_TireRotates). The faster the tire spins, the faster the car moves—large $ω$ means large v because $v = r ω$ . Similarly, a larger-radius tire rotating at the same angular velocity, $ω$ , 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.

View original figure (external; image not reproduced in this text edition)

Original figure description: The diagram shows an illustration of the front part of a car. The diagram shows an arrow on the wheel pointing clockwise and labeled omega (angular velocity). There is a green arrow pointing toward the front of the car labeled v (velocity). The radius of the wheel is labeled r. To the right of the wheel is an equation v equals r times omega.

A car moving at a velocity, v, to the right has a tire rotating with angular velocity

ω

. 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 = r ω

, 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.

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_06_01_Vinyl.

View original figure (external; image not reproduced in this text edition)

Original figure description: The figure shows an illustration of a vinyl record with an arrow omega (angular velocity) pointing in a clockwise direction. There are two lines for the radius, marked r, one going from the center up and the other going from the center to the right. There are three flies on the record. One is positioned at the top of the record on the vertical radius. A v (velocity) arrow points to the right. A second fly is half-way around the circumference toward the horizontal radius and an arrow v is pointing tangential to the fly. The third fly is on the circumference at the horizontal radius and an arrow v is pointing down.

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.

Solving Problems Involving Angle of Rotation and Angular Velocity

Measuring Angular Speed

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

One string (1 m long)
One object (two-hole rubber stopper) to tie to the end
One timer

Procedure

Tie an object to the end of a string.
Swing the object around in a horizontal circle above your head (swing from your wrist). It is important that the circle be horizontal!
Maintain the object at uniform speed as it swings.
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 10 revolutions by that time to get the angular speed in revolutions per second, which you can convert to radians per second.
What is the approximate linear speed of the object?
Move your hand up the string so that the length of the string is 90 cm. Repeat steps 2–5.
Move your hand up the string so that its length is 80 cm. Repeat steps 2–5.
Move your hand up the string so that its length is 70 cm. Repeat steps 2–5.
Move your hand up the string so that its length is 60 cm. Repeat steps 2–5
Move your hand up the string so that its length is 50 cm. Repeat steps 2–5
Make graphs of angular speed vs. radius (i.e. string length) and linear speed vs. radius. Describe what each graph looks like.

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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.

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?

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 $\frac{1}{4} \times 2 π rad = \frac{π}{2}$ (i.e., 90 degrees).

Solution to (b)

Rearranging the equation

Δ θ = \frac{Δ s}{r},

we get

Δ s = r Δ θ .

Inserting the known values gives an arc length of

\begin{matrix} Δ s & = & (1.0 m) (\frac{π}{2} rad) \\ = & 1.6 m \end{matrix}

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.

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 Figure_06_01_TireRotates.

Solution

To find the angular speed, we use the relationship: $ω = \frac{v}{r}$ .

Inserting the known quantities gives

\begin{matrix} ω & = & \frac{15.0 m/s}{0.300 m} \\ = & 50.0 rad/s. \end{matrix}

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

\begin{matrix} ω & = & \frac{15.0 m/s}{1.20 m} \\ = & 12.5 rad/s \end{matrix}

Practice Problems

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

0°
90°
180°
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?

0.1 m
0.2 m
0.3 m
0.6 m

Check Your Understanding

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

The radius of curvature is the radius of a circular path.
The radius of curvature is the diameter of a circular path.
The radius of curvature is the circumference of a circular path.
The radius of curvature is the area of a circular path.

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Section Summary

Circular motion is motion in a circular path.
The angle of rotation $Δ θ$ is defined as the ratio of the arc length to the radius of curvature.
The arc length $Δ s$ is the distance traveled along a circular path and r is the radius of curvature of the circular path.
The angle of rotation $Δ θ$ is measured in units of radians (rad), where $2 π rad = 360 ° = 1$ revolution.
Angular velocity $ω$ is the rate of change of an angle, where a rotation $Δ θ$ occurs in a time $Δ t$ .
The units of angular velocity are radians per second (rad/s).
Tangential speed v and angular speed $ω$ are related by $v = r ω$ , and tangential velocity has units of m/s.
The direction of angular velocity is along the axis of rotation, away (toward) from you for clockwise (counterclockwise) motion.

Key Equations

Angle of rotation	$Δ θ = \frac{Δ s}{r}$
Angular speed:	$ω = \frac{Δ θ}{Δ t}$
Tangential speed:	$v = r ω$

Concept Items

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What kind of motion is called spin?

Spin is rotational motion of an object about an axis parallel to the axis of the object.
Spin is translational motion of an object about an axis parallel to the axis of the object.
Spin is the rotational motion of an object about its center of mass.
Spin is translational motion of an object about its own axis.

Critical Thinking

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Problems

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Test Prep Multiple Choice

What is 1 radian approximately in degrees?

57.3°
360°
π°
2π°

If the following objects are spinning at the same angular velocities, the edge of which one would have the highest speed?

Mini CD
Regular CD
Vinyl record

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Test Prep Short Answer

What is the rotational analog of linear velocity?

Angular displacement
Angular velocity
Angular acceleration
Angular momentum

What is the rotational analog of distance?

Rotational angle
Torque
Angular velocity
Angular momentum

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Test Prep Extended Response

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What happens to tangential velocity as the radius of an object increases provided the angular velocity remains the same?

It increases because tangential velocity is directly proportional to the radius.
It increases because tangential velocity is inversely proportional to the radius.
It decreases because tangential velocity is directly proportional to the radius.
It decreases because tangential velocity is inversely proportional to the radius.

Glossary

angle of rotation: the ratio of the arc length to the radius of curvature of a circular path
angular velocity: ( $ω$ ) the rate of change in the angular position of an object following a circular path
arc length: ( $Δ s$ ) the distance traveled by an object along a circular path
circular motion: the motion of an object along a circular path
radius of curvature: the distance between the center of a circular path and the path
rotational motion: the circular motion of an object about an axis of rotation
spin: rotation about an axis that goes through the center of mass of the object
tangential velocity: the instantaneous linear velocity of an object in circular or rotational motion

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