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Learning Objectives

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

Understand the analogy between angular momentum and linear momentum
Observe the relationship between torque and angular momentum
Apply the law of conservation of angular momentum

The information presented in this section supports the following AP® learning objectives and science practices:

4.D.2.1 The student is able to describe a model of a rotational system and use that model to analyze a situation in which angular momentum changes due to interaction with other objects or systems. (S.P. 1.2, 1.4)
4.D.2.2 The student is able to plan a data collection and analysis strategy to determine the change in angular momentum of a system and relate it to interactions with other objects and systems. (S.P. 2.2)
4.D.3.1 The student is able to use appropriate mathematical routines to calculate values for initial or final angular momentum, or change in angular momentum of a system, or average torque or time during which the torque is exerted in analyzing a situation involving torque and angular momentum. (S.P. 2.2)
4.D.3.2 The student is able to plan a data collection strategy designed to test the relationship between the change in angular momentum of a system and the product of the average torque applied to the system and the time interval during which the torque is exerted. (S.P. 4.1, 4.2)
5.E.1.1 The student is able to make qualitative predictions about the angular momentum of a system for a situation in which there is no net external torque. (S.P. 6.4, 7.2)
5.E.1.2 The student is able to make calculations of quantities related to the angular momentum of a system when the net external torque on the system is zero. (S.P. 2.1, 2.2)
5.E.2.1 The student is able to describe or calculate the angular momentum and rotational inertia of a system in terms of the locations and velocities of objects that make up the system. Students are expected to do qualitative reasoning with compound objects. Students are expected to do calculations with a fixed set of extended objects and point masses. (S.P. 2.2)

Why does Earth keep on spinning? What started it spinning to begin with? And how does an ice skater manage to spin faster and faster simply by pulling her arms in? Why does she not have to exert a torque to spin faster? Questions like these have answers based in angular momentum, the rotational analog to linear momentum.

By now the pattern is clear—every rotational phenomenon has a direct translational analog. It seems quite reasonable, then, to define angular momentum $L$ as

10.90

$L = Iω .$

This equation is an analog to the definition of linear momentum as $p = mv$ . Units for linear momentum are $kg \cdot m /s$ while units for angular momentum are $kg \cdot m^{2} /s$ . As we would expect, an object that has a large moment of inertia $I$ , such as Earth, has a very large angular momentum. An object that has a large angular velocity $ω$ , such as a centrifuge, also has a rather large angular momentum.

Making Connections

Angular momentum is completely analogous to linear momentum, first presented in Uniform Circular Motion and Gravitation. It has the same implications in terms of carrying rotation forward, and it is conserved when the net external torque is zero. Angular momentum, like linear momentum, is also a property of the atoms and subatomic particles.

Example 10.11 Calculating Angular Momentum of the Earth

Strategy

No information is given in the statement of the problem; so we must look up pertinent data before we can calculate $L = Iω$ . First, according to Figure 10.12, the formula for the moment of inertia of a sphere is

10.91

$I = \frac{2 {MR}^{2}}{5}$

so that

10.92

$L = Iω = \frac{2 {MR}^{2} ω}{5} .$

Earth's mass $M$ is $5 . 979 \times 10^{24} kg$ and its radius $R$ is $6 . 376 \times 10^{6} m$ . Earth's angular velocity $ω$ is, of course, exactly one revolution per day, but we must covert $ω$ to radians per second to do the calculation in SI units.

Solution

Substituting known information into the expression for $L$ and converting $ω$ to radians per second gives

10.93

$\begin{array}{l} L & = & 0.4 (5 . 979 \times 10^{24} kg) {(6 . 376 \times 10^{6} m)}^{2} (\frac{1 rev}{d}) \\ = & 9 . 72 \times 10^{37} kg \cdot m^{2} \cdot rev/d . \end{array}$

Substituting $2π$ rad for $1$ rev and $8 . 64 \times 10^{4} s$ for 1 day gives

10.94

$\begin{array}{l} L & = & (9 . 72 \times 10^{37} kg \cdot m^{2}) (\frac{2π rad/rev}{8 . 64 \times 10^{4} s/d}) (1 rev/d) \\ = & 7 . 07 \times 10^{33} kg \cdot m^{2} /s . \end{array}$

Discussion

This number is large, demonstrating that Earth, as expected, has a tremendous angular momentum. The answer is approximate, because we have assumed a constant density for Earth in order to estimate its moment of inertia.

When you push a merry-go-round, spin a bike wheel, or open a door, you exert a torque. If the torque you exert is greater than opposing torques, then the rotation accelerates, and angular momentum increases. The greater the net torque, the more rapid the increase in $L$ . The relationship between torque and angular momentum is

10.95

$net τ = \frac{Δ L}{Δ t} .$

This expression is exactly analogous to the relationship between force and linear momentum, $F = Δ p / Δ t$ . The equation $net τ = \frac{Δ L}{Δ t}$ is very fundamental and broadly applicable. It is, in fact, the rotational form of Newton's second law.

Example 10.12 Calculating the Torque Putting Angular Momentum Into a Lazy Susan

Figure 10.21 shows a llazy susan food tray being rotated by a person in quest of sustenance. Suppose the person exerts a 2.50 N force perpendicular to the lazy susan's 0.260-m radius for 0.150 s. (a) What is the final angular momentum of the lazy susan if it starts from rest, assuming friction is negligible? (b) What is the final angular velocity of the lazy susan, given that its mass is 4 kg and assuming its moment of inertia is that of a disk?

The given figure shows a lazy Susan on which various eatables like cake, salad grapes, and a drink are kept. A hand is shown that applies a force F, indicated by a leftward pointing horizontal arrow. This force is perpendicular to the radius r and thus tangential to the circular lazy Susan.

Figure 10.21 A partygoer exerts a torque on a lazy susan to make it rotate. The equation

$net τ = \frac{Δ L}{Δ t}$ gives the relationship between torque and the angular momentum produced.

Strategy

We can find the angular momentum by solving $net τ = \frac{Δ L}{Δ t}$ for $Δ L$ , and using the given information to calculate the torque. The final angular momentum equals the change in angular momentum, because the lazy susan starts from rest. That is, $Δ L = L$ . To find the final velocity, we must calculate $ω$ from the definition of $L$ in $L = Iω$ .

Solution for (a)

Solving $net τ = \frac{Δ L}{Δ t}$ for $Δ L$ gives

10.96

$Δ L = (net τ) Δt .$

Because the force is perpendicular to $r$ , we see that $net τ = rF$ , so that

10.97

$\begin{array}{l} L & = & rF Δ t = (0 . 260 m) (2.50 N) (0.150 s) \\ = & 9 . 75 \times 10^{- 2} kg \cdot m^{2} / s . \end{array}$

Solution for (b)

The final angular velocity can be calculated from the definition of angular momentum,

10.98

$L = Iω .$

Solving for $ω$ and substituting the formula for the moment of inertia of a disk into the resulting equation gives

10.99

$ω = \frac{L}{I} = \frac{L}{\frac{1}{2} {MR}^{2}} .$

And substituting known values into the preceding equation yields

10.100

$ω = \frac{9 . 75 \times 10^{- 2} kg \cdot m^{2} /s}{(0.500) (4 kg) (0.260 m)} = 0 . 721 rad/s .$

Discussion

Note that the imparted angular momentum does not depend on any property of the object but only on torque and time. The final angular velocity is equivalent to one revolution in 8.71 s (determination of the time period is left as an exercise for the reader), which is about right for a lazy susan.

Take-home Experiment

Plan an experiment to analyze changes to a system's angular momentum. Choose a system capable of rotational motion such as a lazy susan or a merry-go-round. Predict how the angular momentum of this system will change when you add an object to the lazy susan or jump onto the merry-go-round. What variables can you control? What are you measuring? In other words, what are your independent and dependent variables? Are there any independent variables that it would be useful to keep constant. Angular velocity, perhaps? Collect data in order to calculate or estimate the angular momentum of your system when in motion. What do you observe? Collect data in order to calculate the change in angular momentum as a result of the interaction you performed.

Using your data, how does the angular momentum vary with the size and location of an object added to the rotating system?

Example 10.13 Calculating the Torque in a Kick

The person whose leg is shown in Figure 10.22 kicks his leg by exerting a 2,000N force with his upper leg muscle. The effective perpendicular lever arm is 2.20 cm. Given the moment of inertia of the lower leg is $1.25 kg \cdot m^{2}$ , (a) find the angular acceleration of the leg. (b) Neglecting the gravitational force, what is the rotational kinetic energy of the leg after it has rotated through $57 . 3º$ or (1 rad)?

The figure shows a human leg, from the thighs to the feet which is bent at the knee joint. The radius of curvature of the knee is indicated as r equal to two point two zero centimeters and the moment of inertia of the lower half of the leg is indicated as I equal to one point two five kilogram meter square. The direction of torque is indicated by a red arrow in anti-clockwise direction, near the knee.

Figure 10.22 The muscle in the upper leg gives the lower leg an angular acceleration and imparts rotational kinetic energy to it by exerting a torque about the knee.

$F$ is a vector that is perpendicular to

$r$ . This example examines the situation.

Strategy

The angular acceleration can be found using the rotational analog to Newton's second law, or $α = net τ / I$ . The moment of inertia $I$ is given and the torque can be found easily from the given force and perpendicular lever arm. Once the angular acceleration $α$ is known, the final angular velocity and rotational kinetic energy can be calculated.

Solution to (a)

From the rotational analog to Newton's second law, the angular acceleration $α$ is

10.101

$α = \frac{net τ}{I} .$

Because the force and the perpendicular lever arm are given and the leg is vertical so that its weight does not create a torque, the net torque is thus

10.102

$\begin{array}{l} net t & = & r_{?} F \\ = & (0 . 0220 m) (2,000 N) \\ = & 44 m. \end{array}$

Substituting this value for the torque and the given value for the moment of inertia into the expression for $α$ gives

10.103

$α = \frac{44 N \cdot m}{1.25 kg \cdot m^{2}} = 35 . 2 {rad/s}^{2} .$

Solution to (b)

The final angular velocity can be calculated from the kinematic expression

10.104

$ω^{2} = {ω_{0}}^{2} + 2 αθ$

10.105

$ω^{2} = 2 αθ$

because the initial angular velocity is zero. The kinetic energy of rotation is

10.106

${KE}_{rot} = \frac{1}{2} {Iω}^{2}$

so it is most convenient to use the value of $ω^{2}$ just found and the given value for the moment of inertia. The kinetic energy is then

10.107

$\begin{array}{l} {KE}_{rot} & = & 0.5 (1 .25 kg \cdot m^{2}) (70. 4 {rad}^{2} / s^{2}) \\ = & 44 J. \end{array}$

Discussion

These values are reasonable for a person kicking his leg starting from the position shown. The weight of the leg can be neglected in part (a) because it exerts no torque when the center of gravity of the lower leg is directly beneath the pivot in the knee. In part (b), the force exerted by the upper leg is so large that its torque is much greater than that created by the weight of the lower leg as it rotates. The rotational kinetic energy given to the lower leg is enough that it could give a ball a significant velocity by transferring some of this energy in a kick.

Making Connections: Conservation Laws

Angular momentum, like energy and linear momentum, is conserved. This universally applicable law is another sign of underlying unity in physical laws. Angular momentum is conserved when net external torque is zero, just as linear momentum is conserved when the net external force is zero.

Conservation of Angular Momentum

We can now understand why Earth keeps on spinning. As we saw in the previous example, $Δ L = (net τ) Δ t$ . This equation means that, to change angular momentum, a torque must act over some period of time. Because Earth has a large angular momentum, a large torque acting over a long time is needed to change its rate of spin. So what external torques are there? Tidal friction exerts torque that is slowing Earth's rotation, but tens of millions of years must pass before the change is very significant. Recent research indicates the length of the day was 18 h some 900 million years ago. Only the tides exert significant retarding torques on Earth, and so it will continue to spin, although ever more slowly, for many billions of years.

What we have here is, in fact, another conservation law. If the net torque is zero, then angular momentum is constant or conserved. We can see this rigorously by considering $net τ = \frac{Δ L}{Δ t}$ for the situation in which the net torque is zero. In that case,

10.108

$net τ = 0$

implying that

10.109

$\frac{Δ L}{Δ t} = 0 .$

If the change in angular momentum $Δ L$ is zero, then the angular momentum is constant; thus,

10.110

$L = constant (net τ = 0)$

10.111

$L = L' (net τ = 0) .$

These expressions are the law of conservation of angular momentum. Conservation laws are as scarce as they are important.

An example of conservation of angular momentum is seen in Figure 10.23, in which an ice skater is executing a spin. The net torque on her is very close to zero, because there is relatively little friction between her skates and the ice and because the friction is exerted very close to the pivot point. Both $F$ and $r$ are small, and so $τ$ is negligibly small. Consequently, she can spin for quite some time. She can do something else, too. She can increase her rate of spin by pulling her arms and legs in. Why does pulling her arms and legs in increase her rate of spin? The answer is that her angular momentum is constant, so that

10.112

$L = L' .$

Expressing this equation in terms of the moment of inertia,

10.113

$Iω = I' ω',$

where the primed quantities refer to conditions after she has pulled in her arms and reduced her moment of inertia. Because $I'$ is smaller, the angular velocity $ω'$ must increase to keep the angular momentum constant. The change can be dramatic, as the following example shows.

The image a shows an ice skater spinning on the tip of her skate with both her arms and one leg extended. The image b shows the ice skater spinning on the tip of one skate, with her arms crossed and one leg supported on another.

Figure 10.23 (a) An ice skater is spinning on the tip of her skate with her arms extended. Her angular momentum is conserved because the net torque on her is negligibly small. In the next image, her rate of spin increases greatly when she pulls in her arms, decreasing her moment of inertia. The work she does to pull in her arms results in an increase in rotational kinetic energy.

Example 10.14 Calculating the Angular Momentum of a Spinning Skater

Suppose an ice skater, such as the one in Figure 10.23, is spinning at 0.800 rev/ s with her arms extended. She has a moment of inertia of $2.34 kg \cdot m^{2}$ with her arms extended and of $0.363 kg \cdot m^{2}$ with her arms close to her body. (These moments of inertia are based on reasonable assumptions about a 60.0-kg skater.) (a) What is her angular velocity in revolutions per second after she pulls in her arms? (b) What is her rotational kinetic energy before and after she does this?

Strategy

In the first part of the problem, we are looking for the skater's angular velocity $ω'$ after she has pulled in her arms. To find this quantity, we use the conservation of angular momentum and note that the moments of inertia and initial angular velocity are given. To find the initial and final kinetic energies, we use the definition of rotational kinetic energy given by

10.114

${KE}_{rot} = \frac{1}{2} {Iω}^{2} .$

Solution for (a)

Because torque is negligible (as discussed above), the conservation of angular momentum given in $Iω = I' ω'$ is applicable. Thus,

10.115

$L = L'$

10.116

$Iω = I' ω'$

Solving for $ω'$ and substituting known values into the resulting equation gives

10.117

$\begin{array}{l} ω' & = & \frac{I}{I'} ω = (\frac{2.34 kg \cdot m^{2}}{0 .363 kg \cdot m^{2}}) (0.800 rev/s) \\ = & 5.16 rev/s. \end{array}$

Solution for (b)

Rotational kinetic energy is given by

10.118

${KE}_{rot} = \frac{1}{2} {Iω}^{2} .$

The initial value is found by substituting known values into the equation and converting the angular velocity to rad/s:

10.119

$\begin{array}{l} {KE}_{rot} & = & (0 . 5) (2.34 kg \cdot m^{2}) {((0.800 rev/s) (2π rad/rev))}^{2} \\ = & 29.6 J. \end{array}$

The final rotational kinetic energy is

10.120

${KE}_{rot}' = \frac{1}{2} I' {ω'}^{2} .$

Substituting known values into this equation gives

10.121

$\begin{array}{l} K E_{rot}' & = & (0.5) (0 .363 kg \cdot m^{2}) {[(5 . 16 rev/s) (2π rad/rev)]}^{2} \\ = & 191 J. \end{array}$

Discussion

In both parts, there is an impressive increase. First, the final angular velocity is large, although most world-class skaters can achieve spin rates about this great. Second, the final kinetic energy is much greater than the initial kinetic energy. The increase in rotational kinetic energy comes from work done by the skater in pulling in her arms. This work is internal work that depletes some of the skater's food energy.

There are several other examples of objects that increase their rate of spin because something reduced their moment of inertia. Tornadoes are one example. Storm systems that create tornadoes are slowly rotating. When the radius of rotation narrows, even in a local region, angular velocity increases, sometimes to the furious level of a tornado. Earth is another example. Our planet was born from a huge cloud of gas and dust, the rotation of which came from turbulence in an even larger cloud. Gravitational forces caused the cloud to contract, and the rotation rate increased as a result. (See Figure 10.24.)

The first figure shows a cloud of dust and gas,which is in the shape of a distorted circle, rotating in anti-clockwise direction with an angular velocity omega, indicated by a curved black arrow, and having an angular momentum L. The second figure shows an elliptical cloud of dust with the Sun in the middle of it, rotating in anti-clockwise direction with an angular velocity omega dash, indicated by a curved black arrow, and having an angular momentum L. The third figure depicts the Solar System, with the

Figure 10.24 The Solar System coalesced from a cloud of gas and dust that was originally rotating. The orbital motions and spins of the planets are in the same direction as the original spin and conserve the angular momentum of the parent cloud.

In case of human motion, one would not expect angular momentum to be conserved when a body interacts with the environment as its foot pushes off the ground. Astronauts floating in space aboard the International Space Station have no angular momentum relative to the inside of the ship if they are motionless. Their bodies will continue to have this zero value no matter how they twist about as long as they do not give themselves a push off the side of the vessel.

Check Your Understanding

Is angular momentum completely analogous to linear momentum? What, if any, are their differences?

Solution

Yes, angular and linear momentums are completely analogous. While they are exact analogs they have different units and are not directly inter-convertible like forms of energy are.