ch10 – circular motion ch11 – rotational equilibrium ch12 – rotational motion conceptual...

Ch10 – Circular MotionCh11 – Rotational Equilibrium

Ch12 – Rotational MotionConceptual Physics

Ch10 Frames 2-47 (Asmt:38-47)Ch11 Frames 48-111 (Asmt:98-111)

Ch11 Frames 112-158 (Asmt:147-158)

Centripetal force keeps an object in circular motion.

Two types of circular motion are rotation and revolution.

10.1 Rotation and Revolution

An axis is the straight line around which rotation takes place.

• When an object turns about an internal axis—that is, an axis located within the body of the object—the motion is called rotation, or spin.

• When an object turns about an external axis, the motion is called revolution.


The Ferris wheel turns about an axis. The Ferris wheel rotates, while the riders revolve about its axis.


Earth undergoes both types of rotational motion.

• It revolves around the sun once every 365 ¼ days.

• It rotates around an axis passing through its geographical poles once every 24 hours.


Tangential speed depends on rotational speed and the distance from the axis of rotation.

10.2 Rotational Speed

The turntable rotates around its axis while a ladybug sitting at its edge revolves around the same axis. Which part of the turntable moves faster—the outer part where the ladybug sits or a part near the orange center?

It depends on whether you are talking about linear speed or rotational speed.


Types of SpeedLinear speed is the distance traveled per unit of time.

• A point on the outer edge of the turntable travels a greater distance in one rotation than a point near the center.

• The linear speed is greater on the outer edge of a rotating object than it is closer to the axis.

• The speed of something moving along a circular path can be called tangential speed because the direction of motion is always tangent to the circle.


Rotational speed (sometimes called angular speed) is the number of rotations per unit of time.

• All parts of the rigid turntable rotate about the axis in the same amount of time.

• All parts have the same rate of rotation, or the same number of rotations per unit of time. It is common to express rotational speed in revolutions per minute (RPM).


All parts of the turntable rotate at the same rotational speed.

a. A point farther away from the center travels a longer path in the same time and therefore has a greater tangential speed.

b. A ladybug sitting twice as far from the center moves twice as fast.


In symbol form,v ~ r

where v is tangential speed and (pronounced oh MAY guh) is rotational speed.

• You move faster if the rate of rotation increases (bigger ).

• You also move faster if you are farther from the axis (bigger r).


At the axis of the rotating platform, you have no tangential speed, but you do have rotational speed. You rotate in one place. As you move away from the center, your tangential speed increases while your rotational speed stays the same. Move out twice as far from the center, and you have twice the tangential speed.


think!At an amusement park, you and a friend sit on a large rotating disk. You sit at the edge and have a rotational speed of 4 RPM and a linear speed of 6 m/s. Your friend sits halfway to the center. What is her rotational speed? What is her linear speed?

Answer: Her rotational speed is also 4 RPM, and her linear speed is 3 m/s.


Railroad Train Wheels

How do the wheels of a train stay on the tracks?The train wheels stay on the tracks because their rims are slightly tapered.


The wheels of railroad trains are similarly tapered. This tapered shape is essential on the curves of railroad tracks.•On any curve, the distance along the outer part is longer than the distance along the inner part.•When a vehicle follows a curve, its outer wheels travel faster than its inner wheels. This is not a problem because the wheels roll independent of each other.•For a train, however, pairs of wheels are firmly connected like the pair of fastened cups, so they rotate together.


When a train rounds a curve, the wheels have different linear speeds for the same rotational speed.


think!

Train wheels ride on a pair of tracks. For straight-line motion, both tracks are the same length. But which track is longer for a curve, the one on the outside or the one on the inside of the curve?

Answer: The outer track is longer—just as a circle with a greater radius has a greater circumference.


Velocity involves both speed and direction.• When an object moves in a circle, even at

constant speed, the object still undergoes acceleration because its direction is changing.

• This change in direction is due to a net force (otherwise the object would continue to go in a straight line).

• Any object moving in a circle undergoes an acceleration that is directed to the center of the circle—a centripetal acceleration.

10.3 Centripetal Force

Centripetal means “toward the center.”The force directed toward a fixed center that causes an object to follow a circular path is called a centripetal force.


Examples of Centripetal ForcesIf you whirl a tin can on the end of a string, you must keep pulling on the string—exerting a centripetal force. The string transmits the centripetal force, pulling the can from a straight-line path into a circular path.


The force exerted on a whirling can is toward the center. No outward force acts on the can.

Centripetal force holds a car in a curved path. a. For the car to go around a curve, there must be

sufficient friction to provide the required centripetal force.

b. If the force of friction is not great enough, skidding occurs.


The clothes in a washing machine are forced into a circular path, but the water is not, and it flies off tangentially.


Calculating Centripetal Forces

Greater speed and greater mass require greater centripetal force. Traveling in a circular path with a smaller radius of curvature requires a greater centripetal force.

Centripetal force, Fc, is measured in newtons when m is expressed in kilograms, v in meters/second, and r in meters.

The string of a conical pendulum sweeps out a cone. Only two forces act on the bob: mg, the force due

to gravity, and T, tension in the string. • Both are vectors.

The vector T can be resolved into two perpendicular components, Tx (horizontal), and Ty (vertical). If vector T were replaced with forces represented by these component vectors, the bob would behave just as it does when it is supported only by T.

The vector T can be resolved into a horizontal (Tx) component and a vertical (Ty) component.

The vector T can be resolved into two perpendicular components, Tx (horizontal), and Ty (vertical). If vector T were replaced with forces represented by these component vectors, the bob would behave just as it does when it is supported only by T.

Since the bob doesn’t accelerate vertically, the net force in the vertical direction is zero. Therefore Ty must be equal and opposite to mg. Tx is the net force on the bob–the centripetal force. Its magnitude is mv/r2, where r is the radius of the circular path.


Centripetal force keeps the vehicle in a circular path as it rounds a banked curve.


Suppose the speed of the vehicle is such that the vehicle has no tendency to slide down the curve or up the curve. At that speed, friction plays no role in keeping the vehicle on the track. Only two forces act on the vehicle, one mg, and the other the normal force n (the support force of the surface). Note that n is resolved into nx and ny components.


Sometimes an outward force is also attributed to circular motion. This apparent outward force on a rotating or revolving body is called centrifugal force. Centrifugal means “center-fleeing,” or “away from the center.”

10.4 Centripetal and Centrifugal Forces

When the string breaks, the whirling can moves in a straight line, tangent to—not outward from the center of—its circular path.


In the case of the whirling can, it is a common misconception to state that a centrifugal force pulls outward on the can. In fact, when the string breaks the can goes off in a tangential straight-line path because no force acts on it. So when you swing a tin can in a circular path, there is no force pulling the can outward. Only the force from the string acts on the can to pull the can inward. The outward force is on the string, not on the can.


Centrifugal force is an effect of rotation. It is not part of an interaction and therefore it cannot be a true force.

10.5 Centrifugal Force in a Rotating Reference Frame

From the reference frame of the ladybug inside the whirling can, the ladybug is being held to the bottom of the can by a force that is directed away from the center of circular motion.


•From a stationary frame of reference outside the whirling can, we see there is no centrifugal force acting on the ladybug inside the whirling can. •However, we do see centripetal force acting on the can, producing circular motion.• Nature seen from the reference frame of the rotating system is different. •In the rotating frame of reference of the whirling can, both centripetal force (supplied by the can) and centrifugal force act on the ladybug.


In a rotating reference frame the centrifugal force has no agent such as mass—there is no interaction counterpart. For this reason, physicists refer to centrifugal force as a fictitious force, unlike gravitational, electromagnetic, and nuclear forces. Nevertheless, to observers who are in a rotating system, centrifugal force is very real. Just as gravity is ever present at Earth’s surface, centrifugal force is ever present within a rotating system.


1. Whereas a rotation takes place about an axis that is internal, a revolution takes place about an axis that isa. external.b. at the center of gravity.c. at the center of mass.d. either internal or external.

Assessment Questions

1. Whereas a rotation takes place about an axis that is internal, a revolution takes place about an axis that isa. external.b. at the center of gravity.c. at the center of mass.d. either internal or external.

Answer: A


2. When you roll a tapered cup across a table, the path of the cup curves because the wider end rollsa. slower.b. at the same speed as the narrow part.c. faster.d. in an unexplained way.


2. When you roll a tapered cup across a table, the path of the cup curves because the wider end rollsa. slower.b. at the same speed as the narrow part.c. faster.d. in an unexplained way.

Answer: C


3. When you whirl a tin can in a horizontal circle overhead, the force that holds the can in the path acts a. in an inward direction.b. in an outward direction.c. in either an inward or outward direction.d. parallel to the force of gravity.


3. When you whirl a tin can in a horizontal circle overhead, the force that holds the can in the path acts a. in an inward direction.b. in an outward direction.c. in either an inward or outward direction.d. parallel to the force of gravity.

Answer: A


4. When you whirl a tin can in a horizontal circle overhead, the force that the can exerts on the string acts a. in an inward direction.b. in an outward direction.c. in either an inward or outward direction.d. parallel to the force of gravity.


4. When you whirl a tin can in a horizontal circle overhead, the force that the can exerts on the string acts a. in an inward direction.b. in an outward direction.c. in either an inward or outward direction.d. parallel to the force of gravity.

Answer: B


5. A bug inside a can whirled in a circle feels a force of the can on its feet. This force acts a. in an inward direction.b. in an outward direction.c. in either an inward or outward direction.d. parallel to the force of gravity.


5. A bug inside a can whirled in a circle feels a force of the can on its feet. This force acts a. in an inward direction.b. in an outward direction.c. in either an inward or outward direction.d. parallel to the force of gravity.

Answer: A


An object will remain in rotational equilibrium if its center of mass is above the area of support.

Chapter 11

To make an object turn or rotate, apply a torque.

11.1 Torque

A torque produces rotation.

11.1 Torque

When a perpendicular force is applied, the lever arm is the distance between the doorknob and the edge with the hinges.

11.1 Torque

When the force is perpendicular, the distance from the turning axis to the point of contact is called the lever arm. If the force is not at right angle to the lever arm, then only the perpendicular component of the force will contribute to the torque.

11.1 Torque

T = Fd units: nm

Although the magnitudes of the applied forces are the same in each case, the torques are different.

11.1 Torque

When balanced torques act on an object, there is no change in rotation.

11.2 Balanced Torques

A pair of torques can balance each other. Balance is achieved if the torque that tends to produce clockwise rotation by the boy equals the torque that tends to produce counterclockwise rotation by the girl.


do the math!What is the weight of the block hung at the 10-cm mark?


do the math!The block of unknown weight tends to rotate the system of blocks and stick counterclockwise, and the 20-N block tends to rotate the system clockwise. The system is in balance when the two torques are equal:

counterclockwise torque = clockwise torque


do the math!

Rearrange the equation to solve for the unknown weight:The lever arm for the unknown weight is 40 cm. The lever arm for the 20-N block is 30 cm.

The unknown weight is thus 15 N.


The center of mass of an object is the point located at the object’s average position of mass.

11.3 Center of Mass

A baseball thrown into the air follows a smooth parabolic path. A baseball bat thrown into the air does not follow a smooth path. The bat wobbles about a special point. This point stays on a parabolic path, even though the rest of the bat does not.The motion of the bat is the sum of two motions:

• a spin around this point, and • a movement through the air as if all the mass

were concentrated at this point.This point, called the center of mass, is where all the mass of an object can be considered to be concentrated.

11.3 Center of Mass

The centers of mass of the baseball and of the spinning baseball bat each follow parabolic paths.

11.3 Center of Mass

Location of the Center of Mass

For a symmetrical object, such as a baseball, the center of mass is at the geometric center of the object. For an irregularly shaped object, such as a baseball bat, the center of mass is toward the heavier end.

11.3 Center of Mass

The center of mass for each object is shown by the red dot.

11.3 Center of Mass

The center of mass of the toy is below its geometric center.

11.3 Center of Mass

The center of mass of the rotating wrench follows a straight-line path as it slides across a smooth surface.

11.3 Center of Mass

If the wrench were tossed into the air, its center of mass would follow a smooth parabola.

Applying Spin to an Object

When you throw a ball and apply spin to it, or when you launch a plastic flying disk, a force must be applied to the edge of the object. This produces a torque that adds rotation to the projectile.A skilled pool player strikes the cue ball below its center to put backspin on the ball.

11.3 Center of Mass

11.3 Center of Mass

A force must be applied to the edge of an object for it to spin. a. If the football is kicked in line with its

center, it will move without rotating.b. If it is kicked above or below its center, it

will rotate.

For everyday objects, the center of gravity is the same as the center of mass.

11.4 Center of Gravity

Center of mass is often called center of gravity, the average position of all the particles of weight that make up an object. For almost all objects on and near Earth, these terms are interchangeable. There can be a small difference between center of gravity and center of mass when an object is large enough for gravity to vary from one part to another. The center of gravity of the Sears Tower in Chicago is about 1 mm below its center of mass because the lower stories are pulled a little more strongly by Earth’s gravity than the upper stories.


If all the planets were lined up on one side of the sun, the center of gravity of the solar system would lie outside the sun.


Locating the Center of Gravity

The center of gravity (COG) of a uniform object is at the midpoint, its geometric center.

• The COG is the balance point. • Supporting that single point supports the

whole object.


If you suspend any object at a single point, the COG of the object will hang directly below (or at) the point of suspension. To locate an object’s CG:•Construct a vertical line beneath the point of suspension. •The COG lies somewhere alongthat line. •Suspend the object from some other point and construct a second vertical line. •The COG is where the two lines intersect.•You can use a plumb bob to find theCOG for an irregularly shaped object.

There is no material at the COG of these objects.


If the center of gravity of an object is above the area of support, the object will remain upright.

11.5 Torque and Center of Gravity

The block topples when the COG extends beyond its support base.


•The Leaning Tower of Pisa does not topple because its COG does not extend beyond its base. •A vertical line below the COG falls inside the base, and so the Leaning Tower has stood for centuries. •If the tower leaned far enough that the COG extended beyond the base, an unbalanced torque would topple the tower.•The Leaning Tower of Pisa does not topple over because its COG lies above its base.

The Leaning Tower of Pisa does not topple over because its COG lies above its base.


The Moon’s COG

Only one side of the moon continually faces Earth.Because the side of the moon nearest Earth is gravitationally tugged toward Earth a bit more than farther parts, the moon’s COG is closer to Earth than its center of mass. •While the moon rotates about its center of mass, Earth pulls on its COG. •This produces a torque when the moon’s COG is not on the line between the moon’s and Earth’s centers.•This torque keeps one hemisphere of the moon facing Earth.

The center of gravity of a person is not located in a fixed place, but depends on body orientation.

11.6 Center of Gravity of People

When you stand erect with your arms hanging at your sides, your COG is within your body, typically 2 to 3 cm below your navel, and midway between your front and back. Raise your arms vertically overhead. Your COG rises 5 to 8 cm. Bend your body into a U or C shape and your COG may be located outside your body altogether.


A high jumper executes a “Fosbury flop” to clear the bar while his CG nearly passes beneath the bar.


When you stand, your COG is somewhere above your support base, the area bounded by your feet.

• In unstable situations, as in standing in the aisle of a bumpy-riding bus, you place your feet farther apart to increase this area.

• Standing on one foot greatly decreases this area.

• In learning to walk, a baby must learn to coordinate and position the COG above a supporting foot.


You can probably bend over and touch your toes without bending your knees. In doing so, you unconsciously extend the lower part of your body so that your COG, which is now outside your body, is still above your supporting feet. Try it while standing with your heels to a wall. You are unable to adjust your body, and your COG protrudes beyond your feet. You are off balance and torque topples you over.


You can lean over and touch your toes without toppling only if your COG is above the area bounded by your feet.


think!When you carry a heavy load—such as a pail of water—with one arm, why do you tend to hold your free arm out horizontally?

Answer: You tend to hold your free arm outstretched to shift the COG of your body away from the load so your combined COG will more easily be above the base of support. To really help matters, divide the load in two if possible, and carry half in each hand. Or, carry the load on your head!


It is nearly impossible to balance a pen upright on its point, while it is rather easy to stand it upright on its flat end.

• The base of support is inadequate for the point and adequate for the flat end.

• Also, even if you position the pen so that its COG is exactly above its tip, the slightest vibration or air current can cause it to topple.

11.7 Stability

Change in the Location of the COG Upon Toppling

What happens to the COG of a cone standing on its point when it topples? The COG is lowered by any movement. We say that an object balanced so that any displacement lowers its center of mass is in unstable equilibrium.

11.7 Stability

A cone balances easily on its base. To make it topple, its COG must be raised. This means the cone’s potential energy must be increased, which requires work. We say an object that is balanced so that any displacement raises its center of mass is in stable equilibrium.

11.7 Stability

A cone on lying on its side is balanced so that any small movement neither raises nor lowers its center of gravity.The cone is in neutral equilibrium.

11.7 Stability

11.7 Stability

a. Equilibrium is unstable when the COG is lowered with displacement.

b. Equilibrium is stable when work must be done to raise the COG.

c. Equilibrium is neutral when displacement neither raises nor lowers the COG.

Objects in Stable Equilibrium

The horizontally balanced pencil is in unstable equilibrium. Its COG is lowered when it tilts. But suspend a potato from each end and the pencil becomes stable because the COG is below the point of support, and is raised when the pencil is tilted.

11.7 Stability

11.7 Stability

A pencil balanced on the edge of a hand is in unstable equilibrium.a. The COG of the pencil is lowered when it tilts.b. When the ends of the pencil are stuck into

long potatoes that hang below, it is stable because its COG rises when it is tipped.

The toy is in stable equilibrium because the CG rises when the toy tilts.

11.7 Stability

The COG of a building is lowered if much of the structure is below ground level. This is important for tall, narrow structures.

11.7 Stability

11.7 Stability

The COG of an object has a tendency to take the lowest position available.a. A table tennis ball is placed at the bottom

of a container of dried beans.b. When the container is shaken from side

to side, the ball is nudged to the top.

11.7 Stability

The COG of the glass of water is affected by the position of the table tennis ball.a. The COG is higher when the ball is

anchored to the bottom. b. The COG is lower when the ball floats.

1. Applying a longer lever arm to an object so it will rotate producesa. less torque.b. more torque.c. less acceleration.d. more acceleration.


1. Applying a longer lever arm to an object so it will rotate producesa. less torque.b. more torque.c. less acceleration.d. more acceleration.

Answer: B


2. When two children of different weights balance on a seesaw, they each producea. equal torques in the same direction.b. unequal torques.c. equal torques in opposite directions.d. equal forces.


2. When two children of different weights balance on a seesaw, they each producea. equal torques in the same direction.b. unequal torques.c. equal torques in opposite directions.d. equal forces.

Answer: C


3. The center of mass of a donut is locateda. in the hole.b. in material making up the donut.c. near the center of gravity.d. over a point of support.


3. The center of mass of a donut is locateda. in the hole.b. in material making up the donut.c. near the center of gravity.d. over a point of support.

Answer: A


4. The center of gravity of an objecta. lies inside the object.b. lies outside the object.c. may or may not lie inside the object.d. is near the center of mass.


4. The center of gravity of an objecta. lies inside the object.b. lies outside the object.c. may or may not lie inside the object.d. is near the center of mass.

Answer: C


5. An unsupported object will topple over when its center of gravitya. lies outside the object.b. extends beyond the support base.c. is displaced from its center of mass.d. lowers at the point of tipping.


5. An unsupported object will topple over when its center of gravitya. lies outside the object.b. extends beyond the support base.c. is displaced from its center of mass.d. lowers at the point of tipping.

Answer: B


6. The center of gravity of your best friend is locateda. near the belly button.b. at different places depending on body

orientation.c. near the center of mass.d. at a fulcrum when rotation occurs.


6. The center of gravity of your best friend is locateda. near the belly button.b. at different places depending on body

orientation.c. near the center of mass.d. at a fulcrum when rotation occurs.

Answer: B


7. When a stable object is made to topple over, its center of gravitya. is at first raised.b. is lowered.c. plays a minor role.d. plays no role.


7. When a stable object is made to topple over, its center of gravitya. is at first raised.b. is lowered.c. plays a minor role.d. plays no role.

Answer: A


Rotating objects tend to keep rotating while non-rotating objects tend to remain non-rotating.

Chapter 12

Newton’s first law, the law of inertia, applies to rotating objects.

• An object rotating about an internal axis tends to keep rotating about that axis.

• Rotating objects tend to keep rotating, while non-rotating objects tend to remain non-rotating.

• The resistance of an object to changes in its rotational motion is called rotational inertia (sometimes moment of inertia).

12.1 Rotational Inertia

Just as it takes a force to change the linear state of motion of an object, a torque is required to change the rotational state of motion of an object. In the absence of a net torque, a rotating object keeps rotating, while a non-rotating object stays non-rotating.


Rotational Inertia and Mass

Like inertia in the linear sense, rotational inertia depends on mass, but unlike inertia, rotational inertia depends on the distribution of the mass. The greater the distance between an object’s mass concentration and the axis of rotation, the greater the rotational inertia.


By holding a long pole, the tightrope walker increases his rotational inertia.


The short pendulum will swing back and forth more frequently than the long pendulum.


You bend your legs when you run to reduce their rotational inertia. Bent legs are easier to swing back and forth.


Formulas for Rotational Inertia

When all the mass m of an object is concentrated at the same distance r from a rotational axis, then the rotational inertia is I = mr2. When the mass is more spread out, the rotational inertia is less and the formula is different.


Rotational inertias of various objects are different. (It is not important for you to learn these values, but you can see how they vary with the shape and axis.)


think!

When swinging your leg from your hip, why is the rotational inertia of the leg less when it is bent?

Answer: The rotational inertia of any object is less when its mass is concentrated closer to the axis of rotation. Can you see that a bent leg satisfies this requirement?


The human body has three principal axes of rotation.

12.2 Rotational Inertia and Gymnastics

Longitudinal Axis

Rotational inertia is least about the longitudinal axis, which is the vertical head-to-toe axis, because most of the mass is concentrated along this axis.

• A rotation of your body about your longitudinal axis is the easiest rotation to perform.

• Rotational inertia is increased by simply extending a leg or the arms.


An ice skater rotates around her longitudinal axis when going into a spin.

a.The skater has the least amount of rotational inertia when her arms are tucked in.

b.The rotational inertia when both arms are extended is about three times more than in the tucked position.


c and d. With your leg and arms extended, you can vary your spin rate by as much as six times.


A flip involves rotation about the transverse axis.a. Rotational inertia is least in the tuck position. b. Rotational inertia is 1.5 times greater. c. Rotational inertia is 3 times greater. d. Rotational inertia is 5 times greater than in the

tuck position.


Rotational inertia is greater when the axis is through the hands, such as when doing a somersault on the floor or swinging from a horizontal bar with your body fully extended.


The rotational inertia of a body is with respect to the rotational axis.

a.The gymnast has the greatest rotational inertia when she pivots about the bar.

b.The axis of rotation changes from the bar to a line through her center of gravity when she somersaults in the tuck position.


•Which will roll down an incline with greater acceleration, a hollow cylinder or a solid cylinder of the same mass and radius? •The answer is the cylinder with the smaller rotational inertia because the cylinder with the greater rotational inertia requires more time to get rolling. •Inertia of any kind is a measure of “laziness.” •The cylinder with its mass concentrated farthest from the axis of rotation—the hollow cylinder—has the greater rotational inertia. •The solid cylinder will roll with greater acceleration.

A solid cylinder rolls down an incline faster than a hollow one, whether or not they have the same mass or diameter.

12.3 Rotational Inertia and Rolling

A heavy iron cylinder and a light wooden cylinder, similar in shape, roll down an incline. Which will have more acceleration?Answer: The cylinders have different masses, but the same rotational inertia per mass, so both will accelerate equally down the incline. Their different masses make no difference, just as the acceleration of free fall is not affected by different masses. All objects of the same shape have the same “laziness per mass” ratio.

Would you expect the rotational inertia of a hollow sphere about its center to be greater or less than the rotational inertia of a solid sphere? Defend your answer.Answer: Greater. Just as the value for a hoop’s rotational inertia is greater than a solid cylinder’s, the rotational inertia of a hollow sphere would be greater than that of a same-mass solid sphere for the same reason: the mass of the hollow sphere is farther from the center.

12.3 Rotational Inertia and Rolling

Newton’s first law of inertia for rotating systems states that an object or system of objects will maintain its angular momentum unless acted upon by an unbalanced external torque.

12.4 Angular Momentum

Anything that rotates keeps on rotating until something stops it. Angular momentum is defined as the product of rotational inertia, I, and rotational velocity, .angular momentum = rotational inertia × rotational velocity () L = I ×

12.4 Angular Momentum, L

Like linear momentum, angular momentum is a vector quantity and has direction as well as magnitude.

• When a direction is assigned to rotational speed, we call it rotational velocity.

• Rotational velocity is a vector whose magnitude is the rotational speed.

12.4 Angular Momentum, L

For the case of an object that is small compared with the radial distance to its axis of rotation, the angular momentum is simply equal to the magnitude of its linear momentum, mv, multiplied by the radial distance, r. L= mvrThis applies to a tin can swinging from a long string or a planet orbiting in a circle around the sun.

12.4 Angular Momentum,L

An object of concentrated mass m whirling in a circular path of radius r with a speed v has angular momentum mvr.


It is easier to balance on a moving bicycle than on one at rest.

• The spinning wheels have angular momentum.

• When our center of gravity is not above a point of support, a slight torque is produced.

• When the wheels are at rest, we fall over. • When the bicycle is moving, the wheels

have angular momentum, and a greater torque is required to change the direction of the angular momentum.

The lightweight wheels on racing bikes have less angular momentum than those on recreational bikes, so it takes less effort to get them turning.


Angular momentum is conserved for systems in rotation. The law of conservation of angular momentum states that if no unbalanced external torque acts on a rotating system, the angular momentum of that system is constant. With no external torque, the product of rotational inertia and rotational velocity at one time will be the same as at any other time.

12.5 Conservation of Angular Momentum

When the man pulls his arms and the whirling weights inward, he decreases his rotational inertia, and his rotational speed correspondingly increases.


Although the cat is dropped upside down, it is able to rotate so it can land on its feet.


From within a rotating frame of reference, there seems to be an outwardly directed centrifugal force, which can simulate gravity.

12.6 Simulated Gravity

Challenges of Simulated Gravity

The comfortable 1 g we experience at Earth’s surface is due to gravity. Inside a rotating spaceship the acceleration experienced is the centripetal/centrifugal acceleration due to rotation. The magnitude of this acceleration is directly proportional to the radial distance and the square of the rotational speed. At the axis where radial distance is zero, there is no acceleration due to rotation.


If the structure rotates so that inhabitants on the inside of the outer edge experience 1 g, then halfway between the axis and the outer edge they would experience only 0.5 g.


1. The rotational inertia of an object is greater when most of the mass is locateda. near the center.b. off center.c. on the rotational axis.d. away from the rotational axis.


1. The rotational inertia of an object is greater when most of the mass is locateda. near the center.b. off center.c. on the rotational axis.d. away from the rotational axis.

Answer: D


2. How many principal axes of rotation are found in the human body?a. oneb. twoc. threed. four


2. How many principal axes of rotation are found in the human body?a. oneb. twoc. threed. four

Answer: C


3. For round objects rolling on an incline, the faster objects are generally those with thea. greatest rotational inertia compared with

mass.b. lowest rotational inertia compared with

mass.c. most streamlining.d. highest center of gravity.


3. For round objects rolling on an incline, the faster objects are generally those with thea. greatest rotational inertia compared with

mass.b. lowest rotational inertia compared with

mass.c. most streamlining.d. highest center of gravity.

Answer: B


4. For an object traveling in a circular path, its angular momentum doubles when its linear speeda. doubles and its radius remains the same.b. remains the same and its radius doubles.c. and its radius remain the same and its mass

doubles.d. all of the above


4. For an object traveling in a circular path, its angular momentum doubles when its linear speeda. doubles and its radius remains the same.b. remains the same and its radius doubles.c. and its radius remain the same and its mass

doubles.d. all of the above

Answer: D


5. The angular momentum of a system is conserveda. never.b. at some times.c. at all times.d. when angular velocity remains unchanged.


5. The angular momentum of a system is conserveda. never.b. at some times.c. at all times.d. when angular velocity remains unchanged.

Answer: B


6. Gravity can be simulated for astronauts in outer space if their habitata. is very close to Earth.b. is in free fall about Earth.c. rotates.d. revolves about Earth.


6. Gravity can be simulated for astronauts in outer space if their habitata. is very close to Earth.b. is in free fall about Earth.c. rotates.d. revolves about Earth.

Answer: C


ch10 – circular motion ch11 – rotational equilibrium ch12 – rotational motion conceptual...

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