Rotational Motion

Section Learning Objectives

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

Describe rotational kinematic variables and equations and relate them to their linear counterparts
Describe torque and lever arm
Solve problems involving torque and rotational kinematics

Section Key Terms

angular acceleration	kinematics of rotational motion	lever arm
tangential acceleration	torque

Rotational Kinematics

In the section on uniform circular motion, we discussed motion in a circle at constant speed and, therefore, constant angular velocity. However, there are times when angular velocity is not constant—rotational motion can speed up, slow down, or reverse directions. Angular velocity is not constant when a spinning skater pulls in her arms, when a child pushes a merry-go-round to make it rotate, or when a CD slows to a halt when switched off. In all these cases, angular acceleration occurs because the angular velocity $ω ω$ changes. The faster the change occurs, the greater is the angular acceleration. Angular acceleration $α α$ is the rate of change of angular velocity. In equation form, average angular acceleration is

α = \frac{Δ ω}{Δ t}, α = \frac{Δ ω}{Δ t},

where $Δ ω Δ ω$ is the change in angular velocity and $Δ t Δ t$ is the change in time. The units of angular acceleration are (rad/s)/s, or rad/s². If $ω ω$ increases, then $α α$ is positive. If $ω ω$ decreases, then $α α$ is negative. Keep in mind that, by convention, counterclockwise is the positive direction and clockwise is the negative direction. For example, the skater in Figure 6.9 is rotating counterclockwise as seen from above, so her angular velocity is positive. Acceleration would be negative, for example, when an object that is rotating counterclockwise slows down. It would be positive when an object that is rotating counterclockwise speeds up.

The diagram shows a picture of an ice skater with an arrow pointing around her leg indicating that she is spinning in a counterclockwise direction.

Figure 6.9 A figure skater spins in the counterclockwise direction, so her angular velocity is normally considered to be positive. (Luu, Wikimedia Commons)

The relationship between the magnitudes of tangential acceleration, a, and angular acceleration,

α, is a = r α or α = \frac{a}{r} . α, is a = r α or α = \frac{a}{r} .

6.10

These equations mean that the magnitudes of tangential acceleration and angular acceleration are directly proportional to each other. The greater the angular acceleration, the larger the change in tangential acceleration, and vice versa. For example, consider riders in their pods on a Ferris wheel at rest. A Ferris wheel with greater angular acceleration will give the riders greater tangential acceleration because, as the Ferris wheel increases its rate of spinning, it also increases its tangential velocity. Note that the radius of the spinning object also matters. For example, for a given angular acceleration $α α$ , a smaller Ferris wheel leads to a smaller tangential acceleration for the riders.

Tips For Success

Tangential acceleration is sometimes denoted a_t. It is a linear acceleration in a direction tangent to the circle at the point of interest in circular or rotational motion. Remember that tangential acceleration is parallel to the tangential velocity (either in the same direction or in the opposite direction.) Centripetal acceleration is always perpendicular to the tangential velocity.

So far, we have defined three rotational variables: $θ θ$ , $ω ω$ , and $α α$ . These are the angular versions of the linear variables x, v, and a. The following equations in the table represent the magnitude of the rotational variables and only when the radius is constant and perpendicular to the rotational variable. Table 6.2 shows how they are related.

Rotational	Linear	Relationship
$θ θ$	x	$θ = \frac{x}{r} θ = \frac{x}{r}$
$ω ω$	v	$ω = \frac{v}{r} ω = \frac{v}{r}$
$α α$	a	$α = \frac{a}{r} α = \frac{a}{r}$

Table 6.2 Rotational and Linear Variables

We can now begin to see how rotational quantities like $θ θ$ , $ω ω$ , and $α α$ are related to each other. For example, if a motorcycle wheel that starts at rest has a large angular acceleration for a fairly long time, it ends up spinning rapidly and rotates through many revolutions. Putting this in terms of the variables, if the wheel’s angular acceleration $α α$ is large for a long period of time t, then the final angular velocity $ω ω$ and angle of rotation $θ θ$ are large. In the case of linear motion, if an object starts at rest and undergoes a large linear acceleration, then it has a large final velocity and will have traveled a large distance.

The kinematics of rotational motion describes the relationships between the angle of rotation, angular velocity, angular acceleration, and time. It only describes motion—it does not include any forces or masses that may affect rotation (these are part of dynamics). Recall the kinematics equation for linear motion: $v = v_{0} + a t v = v_{0} + a t$ (constant a).

As in linear kinematics, we assume a is constant, which means that angular acceleration $α α$ is also a constant, because $a = r α a = r α$ . The equation for the kinematics relationship between $ω ω$ , $α α$ , and t is

$ω = ω_{0} + α t (constant α), ω = ω_{0} + α t (constant α),$

where $ω_{0} ω_{0}$ is the initial angular velocity. Notice that the equation is identical to the linear version, except with angular analogs of the linear variables. In fact, all of the linear kinematics equations have rotational analogs, which are given in Table 6.3. These equations can be used to solve rotational or linear kinematics problem in which a and $α α$ are constant.

Rotational	Linear
$θ = \bar{ω} t θ = \bar{ω} t$	$x = \bar{v} t x = \bar{v} t$
$ω = ω_{0} + α t ω = ω_{0} + α t$	$v = v_{0} + a t v = v_{0} + a t$	constant $α α$ , a
$θ = ω_{0} t + \frac{1}{2} α t^{2} θ = ω_{0} t + \frac{1}{2} α t^{2}$	$x = v_{0} t + \frac{1}{2} a t^{2} x = v_{0} t + \frac{1}{2} a t^{2}$	constant $α α$ , a
$ω^{2} = ω_{0}^{2} + 2 α θ ω^{2} = ω_{0} 2 + 2 α θ$	$v^{2} = v_{0}^{2} + 2 a x v^{2} = v_{0} 2 + 2 a x$	constant $α α$ , a

Table 6.3 Equations for Rotational Kinematics

In these equations, $ω_{0} ω_{0}$ and $v_{0} v_{0}$ are initial values, $t_{0} t_{0}$ is zero, and the average angular velocity $\bar{ω} \bar{ω}$ and average velocity $\bar{v} \bar{v}$ are

\bar{ω} = \frac{ω_{0} + ω}{2} and \bar{v} = \frac{v_{0} + v}{2} . \bar{ω} = \frac{ω_{0} + ω}{2} and \bar{v} = \frac{v_{0} + v}{2} .

6.11

Fun In Physics

Storm Chasing

The figure shows a photograph of a dark tornado funnel cloud touching the ground and kicking up a lot of debris.

Figure 6.10 Tornadoes descend from clouds in funnel-like shapes that spin violently. (Daphne Zaras, U.S. National Oceanic and Atmospheric Administration)

Storm chasers tend to fall into one of four groups: Amateurs chasing tornadoes as a hobby, atmospheric scientists gathering data for research, weather watchers for news media, or scientists having fun under the guise of work. Storm chasing is a dangerous pastime because tornadoes can change course rapidly with little warning. Since storm chasers follow in the wake of the destruction left by tornadoes, changing flat tires due to debris left on the highway is common. The most active part of the world for tornadoes, called tornado alley, is in the central United States, between the Rocky Mountains and Appalachian Mountains.

Tornadoes are perfect examples of rotational motion in action in nature. They come out of severe thunderstorms called supercells, which have a column of air rotating around a horizontal axis, usually about four miles across. The difference in wind speeds between the strong cold winds higher up in the atmosphere in the jet stream and weaker winds traveling north from the Gulf of Mexico causes the axis of the column of rotating air to shift as the storm travels so that the axis becomes vertical, creating a tornado.

Tornadoes produce wind speeds as high as 500 km/h (approximately 300 miles/h), particularly at the bottom where the funnel is narrowest because the rate of rotation increases as the radius decreases. They blow houses away as if they were made of paper and have been known to pierce tree trunks with pieces of straw.

Torque

If you have ever spun a bike wheel or pushed a merry-go-round, you know that force is needed to change angular velocity. The farther the force is applied from the pivot point (or fulcrum), the greater the angular acceleration. For example, a door opens slowly if you push too close to its hinge, but opens easily if you push far from the hinges. Furthermore, we know that the more massive the door is, the more slowly it opens; this is because angular acceleration is inversely proportional to mass. These relationships are very similar to the relationships between force, mass, and acceleration from Newton’s second law of motion. Since we have already covered the angular versions of distance, velocity and time, you may wonder what the angular version of force is, and how it relates to linear force.

The angular version of force is torque $τ τ$ , which is the turning effectiveness of a force. See Figure 6.11. The equation for the magnitude of torque is

τ = r F \sin θ, τ = r F \sin θ,

where r is the distance from the pivot point to the point where the force is applied, F is the magnitude of the linear force, and $θ θ$ is the angle between the lever arm and the force. The lever arm is the vector from the point of rotation (pivot point or fulcrum) to the location where force is applied. Since the magnitude of the lever arm is a distance, its units are in meters, and torque has units of N⋅m. Torque is a vector quantity and has the same direction as the angular acceleration that it produces.

The figure shows an illustration of a merry-go-round with a radius of r and a man pushing on a bar on the circumference with an arrow indicating a force of F. To the right of the diagram, it is noted that F is perpendicular to r for maximum alpha.

Figure 6.11 A man pushes a merry-go-round at its edge and perpendicular to the lever arm to achieve maximum torque.

Applying a stronger torque will produce a greater angular acceleration. For example, the harder the man pushes the merry-go-round in Figure 6.11, the faster it accelerates. Furthermore, the more massive the merry-go-round is, the slower it accelerates for the same torque. If the man wants to maximize the effect of his force on the merry-go-round, he should push as far from the center as possible to get the largest lever arm and, therefore, the greatest torque and angular acceleration. Torque is also maximized when the force is applied perpendicular to the lever arm.

Solving Rotational Kinematics and Torque Problems

Just as linear forces can balance to produce zero net force and no linear acceleration, the same is true of rotational motion. When two torques of equal magnitude act in opposing directions, there is no net torque and no angular acceleration, as you can see in the following video. If zero net torque acts on a system spinning at a constant angular velocity, the system will continue to spin at the same angular velocity.

Watch Physics

Introduction to Torque

This video defines torque in terms of moment arm (which is the same as lever arm). It also covers a problem with forces acting in opposing directions about a pivot point. (At this stage, you can ignore Sal’s references to work and mechanical advantage.)

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