1. Rotation

2. Introduction
A well-thrown football follows a projectile path while spinning around its long axis.
The motion of the football is easy to analyze because the spin does not affect the path and the path does not affect the spin.
That is, the rotational motion about the axis is independent of the motion through the air.
In this chapter we examine rotational motion without examining any accompanying translational motion.

3. Rotational Motion
The rules for rotational motion have many analogies with translational motion.
The distance traveled is no longer measured in ordinary distance units such as feet or meters, but in an angular measure such as degrees, revolutions, or radians.
Just as feet and meters can be converted to each other, these angular measures are related by simple conversion factors.

4. Rotational Motion
There are 360 degrees in a complete circle, so 1 degree is equal to revolutions.
You may not have encountered the radian:
it is defined as the angle for which the arc length along the circle is equal to the radius of the circle.
The figure shows the size of the radian. There are radians in a complete circle, so a radian is a little larger than 57 degrees.

5. Rotational Motion
Just as translational speed v is the displacement Dx (change in position) divided by the time required, rotational speed w is the angular displacement Dq (change in angular position) divided by the time required.
The units used to express this measurement could be radians per second or revolutions per minute (rpm).
A modern compact disc (CD) spins at variable rates between 500 and 200 revolutions per minute as it plays from the inside track to the outside track.

6. Rotational Motion
Rotational acceleration a is a measure of the rate at which the rotational speed changes.
The change in the rotational speed is equal to the final rotational speed minus the initial rotational speed.
If a CD contains 60 minutes of music, its average rotational acceleration is
Because the CD is slowing, this change is negative.
In this example the CD’s rotational speed decreases by 5 revolutions per minute for each minute of music.

7. Rotational Motion
Like translational velocity and acceleration, rotational velocity and rotational acceleration are vectors.
The assignment of directions to these rotational vectors is not as obvious as it is for their translational counterparts.
The only directions associated with a rotating body that are not continually changing are the directions of the axis of rotation.
Therefore, we assign the direction of the rotational velocity to be along the axis of rotation.

8. Rotational Motion
By convention, if you curl the fingers of your right hand along the direction of rotation as shown in the figure, your thumb points along the axis in the direction of the rotational velocity.
This convention is known as the right-hand rule for rotational velocity.

9. Rotational Motion
The direction of the rotational acceleration is also along the axis of rotation.
If the acceleration causes the object to speed up, the direction of the acceleration is the same as that of the velocity.
If the acceleration causes the object to slow down, the acceleration points in the direction opposite to the velocity.

10. On the Bus
Q: What is the rotational velocity (magnitude and direction) of the second hand of a wristwatch?
A: The second hand makes one complete revolution each minute, or 2p radians every 60 seconds. The magnitude of the rotational velocity is
The direction of the rotational velocity would point along the axis of rotation, into your wrist, by the right-hand rule for rotational velocity.

11. Rotational Motion
Because the relationships between the angular displacement Dq, rotational velocity w, and rotational acceleration a are completely analogous to the relationships between linear displacement Dx, linear velocity v, and linear acceleration a, the kinematic equations that we encountered in chapter 2 each have their rotational counterpart:
Just as the linear kinematic equations were valid only for constant acceleration, the rotational kinematic equations can be used only when rotational acceleration is constant.

12. Torque
Newton’s first law has the same form for rotational motion as it does for translational motion.
The first law says that in the absence of a net external interaction, the natural motion is one in which the rotational velocity remains constant.
If the object is not rotating, it continues to not rotate.
If it is rotating, it continues to rotate with the same rotational velocity.

13. Torque This can be seen with the thrown wrench in the figure below.
If you imagine riding along with the wrench, you see that the wrench rotates about the white dot by the same amount between successive flashes:
that is, its rotational speed is constant.

14. Working it Out: Rotational Kinematics
Let’s figure out how many revolutions our CD makes during the 60 minutes of music.
Using the second rotational kinematic equation, we get
We can also express this angular displacement in units of radians,
where we have used the fact that each revolution is 2p radians.

15. Torque
A change in the rotational speed can only occur when there is a net external interaction on the object.
This interaction involves forces, but unlike translational motion, the locations at which the forces act are as important as their sizes and directions.
The same force can produce different effects depending on where and in which direction it is applied.

16. Torque
You can experiment with these ideas by exerting different forces on a door to your room.
If you push directly toward the hinges or pull directly away from the hinges, the door does not rotate.
Rotations only occur when a horizontal force is applied in any other direction:
the largest occurs when the force is perpendicular to the face of the door.

17. Torque
Even when you apply the force in the perpendicular direction, you get different results depending on where you push.
Try opening the door by pushing at different distances from the hinges.
The largest effect occurs when the force is applied farthest from the hinges.
That’s why doorknobs are put there!

18. Torque
The rotational analog of force is called torque and combines the effects of the force on the door and the distance from the hinges.
If you push on the doorknob, the doorknob moves along a circular path with a radius equal to the distance from the hinge to the doorknob.
If we restrict ourselves to the case when the force is perpendicular to the radius, the magnitude of the torque t is equal to the radius r multiplied by the force F:

19. Torque
Although we will find that torque is a vector that also lies along the axis of rotation, it is easier for us to describe a torque by the effect it has in rotating an object that is initially stationary.
If the object rotates clockwise, we say that the torque is clockwise.
If the object rotates counterclockwise, we say that the torque is counterclockwise.

20. Torque
The development of the concept of torque allows us to restate Newton’s first law for rotation:
The rotational velocity of a rigid object remains constant unless acted on by an unbalanced torque. 
For a vector to remain constant, both its magnitude and its direction must remain constant.

21. Torque
Because torque is equal to a product, we can see why the same applied force can produce different torques on an object.
The torque is increased if the force is applied farther from the axis of rotation.
This fact is useful if you have to loosen a stubborn nut.
The biggest torque occurs when you push or pull the wrench at the spot farthest from the nut.
We can make the distance longer (for the really stubborn nuts) by slipping a pipe over the wrench.

22. Torque
Imagine that you have a flat tire and one of the nuts is stuck tight.
Suppose further that your wrench is 0.3 meter long.
How much torque could you apply by stepping on the end of the wrench?
If you weigh 500 newtons (110 pounds), the maximum torque you could apply would be 150 meter-newtons:

23. On the Bus
Q: Suppose that this is not enough but you found a pipe in your car that could be slipped over the wrench, tripling its effective length. What torque could you now apply?
A: Because the torque is a product of the force and the distance, you would get a torque that is three times as large as the original, or 450 newton-meters. 

24. Torque
When there is more than one applied force, situations can arise when the net force is zero but the net torque is not zero.
In other words, a pair of forces that produces no translational acceleration can still produce rotational acceleration.
The two forces on the board are equal in size and opposite in direction, but they do not act along the same line.
The stick accelerates clockwise because the torques about the center of the board are nonzero and act in the same rotational direction.

25. Torque Look at the seesaw below.
Each girl’s weight multiplied by her distance from the pivot point gives the torque that she applies to the board.
If the torques are equal in magnitude and opposite in direction, there will be no rotational acceleration.
Of course, if this were all that happened, playing on a seesaw would be dull. The seesaw’s motion alternates between two rotations—first in one direction, then in the other.
The momentary torque that makes the transition from one direction to the other is provided when a child pushes off the ground with her feet.

26. Torque
Two people with quite different weights can still balance the seesaw.
The lighter person sits farther from the pivot point, as shown.
This equalizes the torques.
The extra distance compensates for the reduced force due to the smaller weight.

27. Rotational Inertia
The rotational acceleration of an object depends on characteristics of the object as well as the net torque that acts on the object.
If you push on the door to a bank vault and a door in your house, you get different rotational accelerations.
In the translational case, the same net force produces different torque produces different rotational accelerations, but now the acceleration depends on more than the object’s mass:
the distribution of the mass is also important.

28. Rotational Inertia
Consider a “dumbbell” arrangement of a meter stick with a 12-kilogram mass taped on each end.
Holding the dumbbell at its center and rotating it back and forth demonstrates convincingly that a large torque is required to give it a substantial rotational acceleration.
This is completely analogous to the translational inertial properties we have encountered.
The rotational analog of inertia is rotational inertia.

29. Rotational Inertia Changing the arrangement gives different results.
If the two masses are moved closer to the center of the meter stick, it is much easier to start and stop the rotation.
The dumbbell has less rotational inertia even though no mass was removed.
Simply changing the distribution of the mass changed the rotational inertia:
it is larger the farther the mass is located from the point of rotation.

30. Rotational Inertia
Newton’s second law for rotational motion is analogous to that for translational motion, Fnet = ma, where
the net torque t replaces the net force F,
the rotational inertia I replaces the mass m, and
the rotational acceleration w replaces the translational acceleration a:
The net torque on an object is equal to its rotational inertia times its rotational acceleration.

31. Rotational Inertia
Just as translational acceleration must always point in the same direction as the net force causing it, rotational acceleration must always point in the same direction as the net torque.
Therefore, the net torque must lie along the axis of rotation.

32. Flawed Reasoning A group of engineers are designing a machine.
At one place in the machine, a large gear is turned on an axle by a motor.
The large gear meshes with a small gear to turn it on its axle.
The engineers are arguing about the torques that the gears exert on each other.

33. Flawed Reasoning
Seth: “The large gear will exert a larger torque on the small gear than the small gear exerts back on the large gear, by virtue of its size.”
Jason: “You are partially right. The large gear does exert the larger torque, but not because of its size. The large gear is the one attached to the motor. It is driving the small gear, so it must be exerting the larger torque.”
Roger: “You are both forgetting Newton’s third law. The force exerted by the small gear on the large gear is equal and opposite to the force exerted by the large gear on the small gear, so the torques they exert on each other must also be equal.”

34. Flawed Reasoning
Jane: “Newton’s third law applies to forces, but not to torques. Even though the forces exerted by the gears on each other must be equal and opposite, the force that the small gear exerts on the large gear is acting farther away from the axle, so the small gear is actually exerting the larger torque!”
Which of these engineers should be the leader of the project?

35. Flawed Reasoning ANSWER We hope that Jane is directing this project.
She understands that Newton’s third law always applies whenever two objects interact, but that equal and opposite forces does not mean equal and opposite torques.
Because a torque is the product of the force and the distance from the axis of rotation, the force acting farther from the axle will produce the larger torque.
This principle is what makes gears useful.
Note that although there is a rotational analog for Newton’s first and second laws, no such analog exists for Newton’s third law.

36. Rotational Inertia
Losing one’s balance on the high wire amounts to gaining a rotation off the wire.
Tightrope walkers increase their rotational inertia by carrying long poles.
Their increased rotational inertia helps them maintain their balance by allowing them more time to react.
We naturally do something like this when we try to keep our balance.
Picture yourself walking on a railroad track.
Where are your arms?

37. Center of Mass
If we mentally shrink any object so that its entire mass is located at a certain point, the translational motion of this new, very compact object would be the same as that of the original object.
Furthermore, if the object is rotating freely, it rotates about this same point.
This point is called the center of mass.
The center-of-mass concept is also useful for examining the effect of gravity on extended objects.
Rather than dealing with an incredibly large number of gravitational forces acting on each part of the object, we treat the object as if the total force (that is, its weight) acts at the center of mass.
By doing this, we can account for the translational and rotational motions of the object.

38. Center of Mass Now we need a way of finding the center of mass.
This can be determined by a mathematical averaging procedure that considers the distribution of the object’s mass.
A certain amount of mass on one side of the object is balanced, or averaged, with some mass on the other side.
But there are easier ways.

39. Center of Mass
Finding the center of mass for a regularly shaped object is fairly simple.
The symmetry of the object tells us that the center of mass must be at the geometric center of the object.
It is interesting to note that there does not have to be any mass at that spot:
a hollow tennis ball’s center of mass is still at its geometric center.

40. On the Bus
Q: Where would you expect the center of mass of a doughnut to be?
A: Because the doughnut is approximately symmetric, its center of mass is near the center of the hole.

41. Center of Mass
Locating the center of mass of an irregularly shaped object is a little more difficult.
However, because the weight can be considered to act at the center of mass, we can locate it with a simple experiment.
Hang the object from some point along its surface so that it is free to swing, as in (a).
The object will come to rest in a position where there is no net torque on it.

42. Center of Mass
At this position the weight acts along a vertical line through the support point.
Therefore, the center of mass is located someplace on this vertical line.
Now suspend the object from another point, establishing a second line.
Because the center of mass must lie on both lines, it must be at the intersection of the two lines (b).

43. Center of Mass
Try to guess the location of the center of mass of your home state.
You can check your guess by taping a map of your state onto a piece of cardboard.
After cutting around the edges of the state, suspend it from several points to locate its center of mass.

44. Flawed Reasoning
Roger finds the center of mass of a baseball bat by balancing the bat on his finger.
He then saws the bat into two pieces at the location of the center of mass.
He expected the masses of the two pieces to be identical because the average location of the mass must have half the mass on one side and half the mass on the other.
But when he held the two pieces, one was obviously heavier than the other.
What mistake did Roger make in his reasoning?

45. Flawed Reasoning
ANSWER The center of mass of the bat is not the average location of its mass.
It is a weighted average, like the calculation of your GPA.
Because the bat balances at the center of mass, the torque exerted by the weight of the fat end about this pivot must balance the torque exerted by the skinny end about this pivot.
Because the mass in the fat end is located closer to the pivot point, the gravitational force acting on it must be greater.
The fat end therefore weighs more than the skinny end.

46. Stability
We can extend our ideas about rotating objects to see why some things tip over easily and others are quite stable.
Picture a child making a tall tower out of toy blocks.
Much to the child’s delight, the tower always tips over.
But why does this happen?
Clearly there are taller structures in the world than this child’s tower.

47. Stability
We answer this by looking at the stability of a one-block tower.
In the figure the left side of the block is slightly above the table.
If we let go in this position, the block’s weight (acting at the center of mass) provides a counter-clockwise torque about the right edge.
The force by the table on the block acts along this edge but produces no torque because it acts at the pivot.
Thus, the net torque is counterclockwise, and the block falls back to its original position.

48. Stability
In the figure the block is tilted far enough that the weight acts to the right of the pivot point.
The weight produces a clockwise torque, and the block falls over.
The block tips over whenever its center of mass is beyond the edge of the base.

49. Stability
As the child’s toy tower gets taller and its center of mass gets higher, the amount the tower has to sway before the center of mass passes beyond the base gets smaller.
We can make the tower more stable by keeping it short, widening its base, or both.
If you get bumped while standing with your feet close together, you begin to fall over.
To stop this, you quickly spread your feet and increase your support base.
Car manufacturers promote superwide wheelbases because this innovation makes the car more stable.

50. Stability
Tightrope walking is difficult because the support base (the wire’s thickness) is so small.
A slight lean to the left or right puts the center of mass past the support point and creates a torque.
The torque produces a rotation in the same direction as the initial lean, making the situation worse.
Such a situation is known as unstable equilibrium.

51. Stability
The most stable arrangement occurs when the center of mass is below the support point, as in the figure.
As the center of mass sways left or right, the torque that is created rotates the object back to the original orientation.
This situation is known as stable equilibrium.

52. Extended Free-Body Diagrams
If a painter weighing 700 newtons stands in the center of a 300-newton painting platform, we can easily argue that the tension in each of the support cables must be 500 newtons.
The total force acting downward on the platform is the gravitational force of 300 newtons and the 500-newton normal push exerted by the painter.
This total downward force of 1000 newtons must be balanced by a total upward force of 1000 newtons supplied by the two cables.
Symmetry demands that each cable supply half the force.

53. Extended Free-Body Diagrams
If the painter moves to the right, this symmetry is broken, and the cable on the right must support more of the 1000 newtons than the cable on the left.
Newton’s first law no longer provides us with enough information to solve for the new tensions.
We must also use Newton’s first law for rotation and the fact that the platform does not have a rotational acceleration.
This means that the torques acting on the platform must be balanced.

54. Extended Free-Body Diagrams
It does not make sense to talk about a torque on an object without specifying the pivot point about which the torque is acting.
In the case of a stationary object such as the painting platform, it is not rotating about any point, so we are free to choose any point as our pivot location and calculate the torques about that point.

55. Extended Free-Body Diagrams
The torque that results when a force is applied depends on where the force is applied relative to the pivot point.
A simple free-body diagram does not contain this information, as all forces are drawn acting on a dot representing the center of mass of the object.
When balancing torques, we must draw an extended free-body diagram that shows the point at which each force is applied to the object.
We are looking for the new tensions in the two support cables.

56. Extended Free-Body Diagrams
These are forces that are acting on the platform, so we need to draw an extended free-body diagram of the platform, indicating where each of the forces is applied.
We have arbitrarily chosen the left end of the platform as the pivot point about which we will balance torque.
This choice of pivot location removes the torque produced by the left cable from our calculation, as its tension force acts at the pivot.

57. Extended Free-Body Diagrams
The gravitational force acting at the center of the platform and the normal force exerted by the painter are acting in a direction that would make the platform rotate clockwise around our chosen pivot point,
while the tension in the right cable is acting in a direction that would make the platform rotate counterclockwise about this same pivot.
We must therefore balance the torque due to the gravitational force and the torque due to the painter’s normal force with the opposing torque due to the tension in the right cable.

58. Extended Free-Body Diagrams
The tension in the right cable (TR) can now be found to be 675 newtons.
We still require that the total force acting upward on the platform add up to 1000 newtons,
so the tension in the left cable must be 325 newtons.

59. Rotational Kinetic Energy
If we drop a yo-yo to the floor, it speeds up as gravitational potential energy is converted to kinetic energy.
If, instead, we hold on to the string while the yo-yo drops, the yo-yo does not speed up as quickly.
Only part of the lost gravitational potential energy has been converted to translational kinetic energy of the center of mass.
Because the total energy is conserved, the rest must have been converted to a new form of energy.
This new form is associated with the spinning motion of the yo-yo about its center of mass and is called rotational kinetic energy.

60. Rotational Kinetic Energy
The rotational kinetic energy of the yo-yo can be calculated by treating the yo-yo as if it were millions of connected pieces.
If we use our formula for linear kinetic energy,
for each of these pieces and add all of the contributions, we find that the total takes a familiar form.
Whereas the linear kinetic energy depends on the linear inertia (mass) and the square of the linear speed, the rotational kinetic energy depends on the rotational inertia I and the square of the rotational speed w:
As for translational kinetic energy, this quantity is not a vector; no direction is associated with rotational kinetic energy.
Because this is an energy, the units for rotational kinetic energy are joules.

61. Rotational Kinetic Energy
Just as the rotational kinetic energy of the yo-yo can be converted to other forms of energy, the rotational kinetic energy stored in flywheels can be used to make automobiles or buses more fuel efficient.
As the vehicle slows to a stop, the translational kinetic energy can be used to spin up a heavy flywheel.
Then, when the light turns green, this spinning flywheel can be used to accelerate the car.
We will find in the next three sections that building a car with a single flywheel could have disastrous effects.
In practice we need to use two identical flywheels spinning in opposite directions at the same speed.

62. Working it Out: Extended Free-Body Diagrams
The left cable on the painting platform is moved from the left edge to a point 0.5 m to the right.
Let’s find the new tensions in the two cables.

63. Working it Out: Extended Free-Body Diagrams
We start again with an extended free-body diagram of the platform.
We can simplify our calculations if we choose the pivot point to be at the location where the left cable is connected.
The torque exerted by the left cable about this pivot location is zero.

64. Working it Out: Extended Free-Body Diagrams
We balance the two clockwise torques with the one counterclockwise torque using the new distances to the pivot.
The tension in the right cable (TR) is now 567 N.
The total upward force must be 1000 N, so the tension in the left cable is now 433 N.

65. Angular Momentum
There is another kind of momentum in which an object orbiting a point has a rotational quantity of motion that is different from linear momentum.
This new quantity is called angular momentum and is represented by the letter L.
The magnitude of the angular momentum in this example is equal to the object’s linear momentum multiplied by the radius r of its circular path:

66. Angular Momentum
A spinning object also has angular momentum because it is really just a large collection of tiny particles, each of which is revolving around the same axis.
The total angular momentum of a spinning object is just the sum of the individual angular momenta of the individual particles.
We find that the angular momentum of the spinning object is equal to the product of its rotational inertia I and its rotational speed w, L = Iw which is analogous to the expression for linear momentum, p = mv,
where the angular momentum L replaces the linear momentum p,
the rotational inertia I replaces the mass m, and
the rotational velocity w replaces the translational velocity v.

67. Conservation of Angular Momentum
Earth, for example, has both types of angular momenta:
the angular momentum due to its annual revolution around the Sun and that due to its daily rotation on its axis.
The angular momentum of a system does not change under certain circumstances.
The law of conservation of angular momentum is analogous to the conservation law for linear momentum.
The difference is that the interaction that changes the angular momentum is a torque rather than a force.
If the net external torque on a system is zero, the total angular momentum of the system does not change.

68. Conservation of Angular Momentum
Note that the net external force need not be zero for angular momentum to be conserved.
A net external force can be acting on the system as long as the force does not produce a torque.
This is the case for projectile motion because the force of gravity can be considered to act at the object’s center of mass.
Therefore, even though a thrown baton follows a projectile path, it continues to spin with the same angular momentum around its center of mass.
There is no net torque on the baton.

69. Conservation of Angular Momentum
In some interesting situations, the angular momentum of a spinning object is conserved but the object changes its rotational speed.
Near the end of a performance, many ice skaters go into a spin.
The spin usually starts out slowly and then gets faster and faster.
This may appear to be a violation of the law of conservation of angular momentum but is in fact a beautiful example of its validity.

70. Conservation of Angular Momentum
Angular momentum is the product of the rotational inertia and the rotational speed and, in the absence of a net torque, remains constant.
Therefore, if the rotational inertia decreases, the rotational speed must increase.
This is exactly what happens.
The skater usually begins with arms extended.
As the arms are drawn in toward the body, the rotational inertia of the body decreases because the mass of the arms is now closer to the axis of rotation.
This requires that the rotational speed increase.
To slow the spin, the skater reverses the procedure by extending the arms to increase the rotational inertia.

71. Conservation of Angular Momentum
The same principle applies to the flips and twists of gymnasts and springboard divers.
The rate of rotation and hence the number of somersaults that can be completed depends on the rotational inertia of the body as well as the angular momentum and height generated during the take-off.
The figure gives the relative values of the rotational inertia for the tuck, pike, and layout positions.
The more compact tuck has the smallest rotational inertia and therefore has the fastest rotational speed.

72. Conservation of Angular Momentum
Cats have the amazing ability to land on their feet regardless of their initial orientation.
Modern strobe photographs have shown that the cat does not acquire a rotation by kicking off.
The cat’s initial angular momentum is zero.
Because the force of gravity acts through the cat’s center of mass, it produces no torque, and the angular momentum remains zero.

73. Conservation of Angular Momentum
The cat rotates by turning the front and hind ends of its body in different directions.
The entire cat has zero angular momentum as long as the angular momenta of the two parts are equal and opposite.
Even though these angular momenta are the same size, the amount of rotation can be different because it depends on the rotational inertia of that part of the body.
The cat adjusts the rotational inertia by retracting and extending its legs.

74. Conservation of Angular Momentum
As Earth moves along its orbit, it is continually being attracted toward the Sun.
Because the gravitational force always acts toward the Sun, there is no net torque affecting Earth’s motion around the Sun.
Therefore, Earth’s orbital angular momentum must be conserved.
As Kepler discovered, Earth’s orbit about the Sun is not a circle but an ellipse.
Thus, Earth is not always the same distance from the Sun.
This means that when Earth is closer to the Sun, its speed must be faster to keep the angular momentum constant.
Similarly, Earth’s speed must be slower when it is farther from the Sun.

75. Conservation of Angular Momentum
The Solar System began as a huge cloud of gas and dust that had a very small rotation as part of its overall motion around the center of the Galaxy.
As it collapsed under its mutual gravitational attraction, it rotated faster and faster in agreement with conservation of angular momentum.
This explains why the planets all revolve around the Sun in the same direction and why the rotation of the Sun itself is also in this direction.

76. On the Bus
Q: Imagine you are executing a running front somersault when you suddenly realize that you are not turning fast enough to make it around to your feet. What can you do?
A: You can tighten your tuck to reduce your rotational inertia. Because angular momentum is conserved, you will rotate faster.

77. Angular Momentum: A Vector
Like linear momentum, angular momentum is a vector quantity.
The conservation of a vector quantity means that both the magnitude and direction are constant.
There are some interesting consequences of conserving the direction of angular momentum.
The direction of the angular momentum is the same as that of the rotational velocity; that is, it lies along the axis of rotation.

78. Angular Momentum: A Vector
One important application of this principle is the use of a gyroscope for guidance in airplanes and spacecraft.
A gyroscope is simply a disk that is rotating rapidly about an axle.
The axle is mounted so that the mounting can be rotated in any direction without exerting a torque on the rotating disk.
Once the gyroscope is rotating, the axle maintains its direction in space no matter the orientation of the spacecraft.

79. Angular Momentum: A Vector
A couple of students who were studying angular momentum decided to play a practical joke on a classmate.
They mounted a heavy flywheel in an old suitcase and gave it a large rotational speed.
They then asked a classmate to carry the suitcase into another room.
When the classmate turned a corner, the bottom of the suitcase quickly rose, almost spraining his arm!
What happened?

80. Angular Momentum: A Vector
The suitcase did not follow the classmate around the corner because the large angular momentum of the flywheel resisted any change in its orientation.
Not only did it resist any change in its orientation, it turned in a different, and unexpected, direction.

81. Angular Momentum: A Vector
A spinning top has angular momentum, but it is not usually constant.
When the top’s center of mass is not directly over the tip, the gravitational force exerts a torque on the top.
If the top were not spinning, this torque would simply cause it to topple over.
But when it is spinning, the torque causes the angular momentum to change its direction, not its magnitude.

82. Angular Momentum: A Vector
The spin axis of the top (and hence its angular momentum) traces out a cone.
We say that the top precesses.
The friction of the top’s contact point with the table produces another torque.
This torque reduces the magnitude of the angular momentum and eventually causes the top to slow down and topple over.

83. Angular Momentum: A Vector
A similar situation exists with Earth. Because Earth’s shape is irregular, the gravitational forces of the Sun and Moon on Earth produce torques on the spinning Earth.
These torques cause Earth’s spin axis to precess.
This precession is very slow but does cause the direction of our North Pole to sweep out a big cone in the sky once every 25,780 years.
Thus, Polaris (the Pole Star) is not always the North Star.
In about 12,000 years the North Pole will point toward the star Vega, and our descendants will call that star the North Star and use it to navigate.

84. On the Bus
Q: Assume that you are at the North Pole holding a rapidly spinning gyroscope that has its angular momentum vector pointing straight up. Which way will the gyroscope point if you transport it to the South Pole without exerting any torques on it?
A: It will point toward the ground. Remember that this is the same direction (directly toward the North Star) as before. You have changed your orientation because your feet must point toward the center of the spherical Earth.

85. Summary
Objects can rotate or revolve around some axis, and this can happen whether they have a fixed or moving axis.
The rotational and translational motions are independent of each other.
The rotation of a free body takes place about its center of mass.
The rules for rotational motion are similar to the rules for translational motion.
The angular displacement is a change in angular position, rotational velocities are angular displacements divided by time, and rotational accelerations are changes in rotational velocities divided by time.

86. Summary
Newton’s first and second laws for rotational motion have the same form as the laws for translational motion.
A change in rotational velocity occurs only when there is a net torque on the object.
The torque t is equal to the radius r multiplied by the perpendicular force F, or t = rF, and has units of meter-newtons.

87. Summary
Rotational inertia depends on the distance of the mass from the axis of rotation as well as on the mass itself.
The stability of an object depends on the torques produced by its weight (acting at the center of mass) and on the supporting forces.
For an object orbiting a point, its angular momentum is defined as the product of its linear momentum and the radius of its circular path, L = mvr.
For a spinning object, its angular momentum is the product of its rotational inertia and its rotational speed, L = Iw.

88. Summary
The angular momentum of a system is conserved if no net external torque acts on the system.
External forces may act on the system as long as these forces do not produce a net torque.
Even though angular momentum is conserved, the rotational speed can change if the rotational inertia changes.
The conservation of a vector quantity means that both its magnitude and direction are constant.
A change in angular momentum can be a change in magnitude or direction or both.
Conservation of angular momentum can be used to analyze problems such as the motion of tops, gymnasts, and cats.