11 Rotational Mechanics An object will remain in rotational equilibrium if its center of mass is...

11 Rotational Mechanics

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


What determines whether an object will rotate when a force acts on it?

Why doesn’t the Leaning Tower of Pisa rotate and topple over?

What maneuvers does a falling cat make to land on its feet?

This chapter is about the factors that affect rotational equilibrium.


How do you make an object turn or rotate?

11.1 Torque


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

11.1 Torque


Every time you open a door, turn on a water faucet, or tighten a nut with a wrench, you exert a turning force.

Torque is produced by this turning force and tends to produce rotational acceleration.

Torque is different from force.• Forces tend to make things accelerate.• Torques produce rotation.

11.1 Torque


A torque produces rotation.

11.1 Torque


A torque is produced when a force is applied with “leverage.”

• You use leverage when you use a claw hammer to pull a nail from a piece of wood.

• The longer the handle of the hammer, the greater the leverage and the easier the task.

• The longer handle of a crowbar provides even more leverage.

11.1 Torque


A torque is used when opening a door. • A doorknob is placed far away from the turning axis

at its hinges to provide more leverage when you push or pull on the doorknob.

• The direction of your applied force is important. In opening a door, you push perpendicular to the plane of the door.

• A perpendicular push or pull gives more rotation for less effort.

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


The same torque can be produced by a large force with a short lever arm, or a small force with a long lever arm.

The same force can produce different amounts of torque.

Greater torques are produced when both the force and lever arm are large.

11.1 Torque


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

11.1 Torque


think!If you cannot exert enough torque to turn a stubborn bolt, would more torque be produced if you fastened a length of rope to the wrench handle as shown?

11.1 Torque


think!If you cannot exert enough torque to turn a stubborn bolt, would more torque be produced if you fastened a length of rope to the wrench handle as shown?

Answer:

No, because the lever arm is the same. To increase the lever arm, a better idea would be to use a pipe that extends upward.

11.1 Torque


What happens when balanced torques act on an object?

11.2 Balanced Torques


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



Children can balance a seesaw even when their weights are not equal.

Weight alone does not produce rotation—torque does.



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.



Scale balances that work with sliding weights are based on balanced torques, not balanced masses. The sliding weights are adjusted until the counterclockwise torque just balances the clockwise torque. We say the scale is in rotational equilibrium.



How is the center of gravity of an everyday object related to its center of mass?

11.3 Center of Gravity


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



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.



Wobbling

If you threw a wrench so that it rotated as it moved through the air, you’d see it wobble about its center of gravity. The center of gravity itself would follow a parabolic path.

The sun itself wobbles off-center.• As the planets orbit the sun, the center of gravity of the

solar system can lie outside the massive sun.• Astronomers look for similar wobbles in nearby stars—

the wobble is an indication of a star with a planetary system.



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 (CG) of a uniform object is at the midpoint, its geometric center.

• The CG is the balance point. • Supporting that single point supports the whole object.



The weight of the entire stick behaves as if it were concentrated at its center. The small vectors represent the force of gravity along the meter stick, which combine into a resultant force that acts at the CG.



The weight of the entire stick behaves as if it were concentrated at its center. The small vectors represent the force of gravity along the meter stick, which combine into a resultant force that acts at the CG.



If you suspend any object at a single point, the CG 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 CG lies somewhere along that line. • Suspend the object from some other point and

construct a second vertical line. • The CG is where the two lines intersect.



You can use a plumb bob to find the CG for an irregularly shaped object.



The CG of an object may be located where no actual material exists.

• The CG of a ring lies at the geometric center where no matter exists.

• The same holds true for a hollow sphere such as a basketball.



There is no material at the CG of these objects.



think!Where is the CG of a donut?



think!Where is the CG of a donut?

Answer:

In the center of the hole!



think!Can an object have more than one CG?



think!Can an object have more than one CG?

Answer:

No. A rigid object has one CG. If it is nonrigid, such as a piece of clay or putty, and is distorted into different shapes, then its CG may change as its shape is changed. Even then, it has one CG for any given shape.



What is the rule for toppling?

11.3 Torque and Center of Gravity


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



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



The Rule for Toppling

If the CG extends outside the area of support, an unbalanced torque exists, and the object will topple.



This “Londoner” double-decker bus is undergoing a tilt test.

So much of the weight of the vehicle is in the lower part that the bus can be tilted beyond 28° without toppling.



The Leaning Tower of Pisa does not topple because its CG does not extend beyond its base.

A vertical line below the CG falls inside the base, and so the Leaning Tower has stood for centuries.

If the tower leaned far enough that the CG extended beyond the base, an unbalanced torque would topple the tower.



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



The support base of an object does not have to be solid.

An object will remain upright if the CG is above its base of support.



The shaded area bounded by the bottom of the chair legs defines the support base of the chair.



Balancing

Try balancing a broom upright on the palm of your hand.

The support base is quite small and relatively far beneath the CG, so it’s difficult to maintain balance for very long.

After some practice, you can do it if you learn to make slight movements of your hand to exactly respond to variations in balance.



Gyroscopes and computer- assisted motors in the self- balancing electric scooter make continual adjustments to keep the combined CGs of Mark, Tenny, and the vehicles above the support base.



The Moon’s CG

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 CG is closer to Earth than its center of mass.

• While the moon rotates about its center of mass, Earth pulls on its CG.

• This produces a torque when the moon’s CG is not on the line between the moon’s and Earth’s centers.

• This torque keeps one hemisphere of the moon facing Earth.



The moon is slightly football-shaped due to Earth’s gravitational pull.



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


In the absence of an external force, the momentum of an object remains unchanged—conservation of momentum. In this chapter we extend the law of momentum conservation to rotation.


How does rotational inertia affect how easily the rotational speed of an object changes?

11.4 Rotational Inertia


The greater the rotational inertia, the more difficult it is to change the rotational speed of an object.



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).



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.



Rotational inertia depends on the distance of mass from the axis of rotation.



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



A long baseball bat held near its thinner end has more rotational inertia than a short bat of the same mass.

• Once moving, it has a greater tendency to keep moving, but it is harder to bring it up to speed.

• Baseball players sometimes “choke up” on a bat to reduce its rotational inertia, which makes it easier to bring up to speed.

A bat held at its end, or a long bat, doesn’t swing as readily.



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



For similar mass distributions, short legs have less rotational inertia than long legs.



The rotational inertia of an object is not necessarily a fixed quantity.

It is greater when the mass within the object is extended from the axis of rotation.



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?



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?



What are the three principal axes of rotation in the human body?

11.5 Rotational Inertia and Gymnastics


The three principal axes of rotation in the human body are the longitudinal axis, the transverse axis, and the medial axis.



The human body can rotate freely about three principal axes of rotation.

Each of these axes is at right angles to the others and passes through the center of gravity.

The rotational inertia of the body differs about each axis.



The human body has three principal axes of rotation.



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.



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.



Transverse Axis

You rotate about your transverse axis when you perform a somersault or a flip.



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.






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.



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.



The rotational inertia of a gymnast is up to 20 times greater when she is swinging in a fully extended position from a horizontal bar than after dismount when she somersaults in the tuck position.

Rotation transfers from one axis to another, from the bar to a line through her center of gravity, and she automatically increases her rate of rotation by up to 20 times.

This is how she is able to complete two or three somersaults before contact with the ground.



Medial Axis

The third axis of rotation for the human body is the front-to-back axis, or medial axis.

This is a less common axis of rotation and is used in executing a cartwheel.



How does Newton’s first law apply to rotating systems?

11.6 Angular Momentum


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.



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 ()

= I ×



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.



Angular momentum depends on rotational velocity and rotational inertia.



The operation of a gyroscope relies on the vector nature of angular momentum.



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.

angular momentum = mvr

This applies to a tin can swinging from a long string or a planet orbiting in a circle around the sun.



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



An external net force is required to change the linear momentum of an object.

An external net torque is required to change the angular momentum of an object.



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.



What happens to angular momentum when no external torque acts on an object?

11.7 Conservation of Angular Momentum


Angular momentum is conserved when no external torque acts on an object.



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.



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



The man stands on a low-friction turntable with weights extended.

• Because of the extended weights his overall rotational inertia is relatively large in this position.

• As he slowly turns, his angular momentum is the product of his rotational inertia and rotational velocity.

• When he pulls the weights inward, his overall rotational inertia is decreased. His rotational speed increases!

• Whenever a rotating body contracts, its rotational speed increases.



Rotational speed is controlled by variations in the body’s rotational inertia as angular momentum is conserved during a forward somersault. This is done by moving some part of the body toward or away from the axis of rotation.



A falling cat is able to execute a twist and land upright even if it has no initial angular momentum.

During the maneuver the total angular momentum remains zero. When it is over, the cat is not turning.

This cat rotates its body through an angle, but does not create continuing rotation, which would violate angular momentum conservation.



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



1. Applying a longer lever arm to an object so it will rotate produces

a. less torque.

b. more torque.

c. less acceleration.

d. more acceleration.

Assessment Questions


1. Applying a longer lever arm to an object so it will rotate produces

a. 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 produce

a. 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 produce

a. 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 located

a. 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 located

a. 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 object

a. 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 object

a. 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 gravity

a. 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 gravity

a. 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 located

a. 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 located

a. 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



1. The rotational inertia of an object is greater when most of the mass is located

a. 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 located

a. 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. one

b. two

c. three

d. four



2. How many principal axes of rotation are found in the human body?

a. one

b. two

c. three

d. four

Answer: C



4. For an object traveling in a circular path, its angular momentum doubles when its linear speed

a. 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 speed

a. 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 conserved

a. never.

b. at some times.

c. at all times.

d. when angular velocity remains unchanged.



5. The angular momentum of a system is conserved

a. never.

b. at some times.

c. at all times.

d. when angular velocity remains unchanged.

Answer: B


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11 Rotational Mechanics An object will remain in rotational equilibrium if its center of mass is...

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