Physics Final Questions

Describe a collision in which all kinetic energy is lost.

All kinetic energy is lost for any completely inelastic collision in which the final velocity is zero. For example, a head on collision between two identical cars traveling the same speed. The initial momentum is zero so the final momentum and the final velocity must be zero and all the kinetic energy that the cars originally had is changed to thermal energy and deformation energy.

Two solid spheres simultaneously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

As seen from Example 8-13, the speed of a sphere rolling down an incline is independent of both its mass and its radius so they have the same speed at the bottom and reach the bottom at the same time. The more massive sphere has twice as much gravitational potential energy a the top of the incline so it has twice as much total kinetic energy at the bottom of the incline.

What keeps a satellite up in its orbit around the Earth?

Because of the satellites tangential velocity it falls around the Earth instead of down to Earth.

If you are riding on a train that speeds past another train moving in the same direction on an adjacent track, it appears that the other train is moving backward. Why?

Because relative to your frame of reference it is moving backwards.

What would your bathroom scale read if you weighed yourself on an inclined plane? Assume the mechanism functions properly, even at an angle.

A bathroom scale reads the magnitude of the normal force with which you push on it. On an inclined plane, the magnitude of the normal force equals the magnitude of the component of your weight perpendicular to the incline. On the incline the scale reading would be less than your weight.

The gravitational force on the Moon due to the Earth is only about half the force on the Moon due to the Sun. Why isn't the Moon pulled away from the Earth?

Because the Sun also pulls on the Earth. On average the Moon is the same distance from the Sun as Earth, so they both have the same radial acceleration due the Sun.

Two cars emerge side by side from a tunnel. Car A is traveling with a speed of 60 km/h and has an acceleration of 40 km/h/minute. Car B has a speed of 40 km/h and has an acceleration of 60 km/h/minute. which car is passing the other as they come out of the tunnel?

Car A is moving faster so it is passing car B since they emerge side by side. (Later, because Car B has a greater acceleration it will catch and pass car A.)

When blood pressure is measured, why must the jacket be held at the level of the heart?

If the blood pressure is measured at a position lower than the heart then the measured blood pressure will be higher than the blood pressure at the heart, due to the effects of gravity on the blood in the blood vessels. If the pressure is measured at a higher position the measured blood pressure will be low for the same reason. To measure the blood pressure at the heart the measurement must be at the same level as the heart.

A woman swimming upstream is not moving with respect to the shore. Is she doing any work? If she stops swimming and merely floats, is work being done on her?

In order to keep in the same place with respect to the shore the woman has to move water with her hands and feet so she is doing work. When she floats the water does work on her until she is floating at the same speed as the water; after that her kinetic energy is not changing so no work is being done by or on her.

Why don't ships made of iron sink?

Iron is more dense than water so a solid ship of iron would sink. But ships aren't solid iron, they have large open spaces filled with air. And air is much less dense than water so the average density of the ship is less than the density of water and the ship floats.

A ladder, leaning against a wall, makes a 60° angle with the ground. When is it more likely to slip: when a person stands on the ladder near the top or near the bottom? Explain.

It is more likely to slip when the person stands near the top since her lever arm will be greater than when she is near the bottom.

Does a car speedometer measure speed, velocity or both?

It measures the instantaneous speed. You would need a compass to measure velocity.

We claim that momentum and angular momentum are conserved. Yet most moving or rotating objects eventually slow down and stop. Explain.

Momentum is only conserved if the net external force on the object is zero and angular momentum is only conserved if the net external torque on the object is zero. For single objects this is almost never the case since external resistive forces such as friction and air resistance act on the objects causing them to slow down and eventually stop.

Suppose you are sitting on a rotating stool holding a 2 kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Neglecting friction and air resistance and assuming that your arms don't move when you drop the masses your angular velocity will stay the same. This is somewhat surprising since it seems that your rotational inertia decreases when you drop the masses. Before you drop the masses the total rotational inertia of the system (you, the stool, and the masses) is your moment of inertia plus the moment of inertia of the stool plus the moment of inertia of the masses. But dropping the masses doesn't change their moment of inertia so your moment of inertia doesn't change unless you change the position of your body. Since your moment of inertia doesn't change your angular velocity doesn't change.Another way to think about this is in terms of angular momentum. At the instant you drop the masses they are moving tangent to the circle and so they have the same angular momentum they had just before you dropped them. So you have the same angular momentum after dropping them as before and therefore the same angular velocity.Finally we can think about his in terms of work and energy. You do no work on the masses when you drop them (since you just let go of them, you don't move them through a distance) so your rotational kinetic energy doesn't change and your angular velocity doesn't change.

Can a centripetal force ever do work on an object? Explain.

No, since a centripetal force is always perpendicular to the displacement.

Could a nonrigid body be described by a single value of the angular velocity ? Explain.

No, since different parts of the body could be rotating at different rates. One example is the solar system; the different planets orbit the Sun with different angular velocities.

Repeat Question 22 for the power needed rather than the work.

Power is the rate of doing work so the power needed depends on the time it takes as well as the other factors.

You pull a box with a constant force across a frictionless table using an attached rope held horizontally. If you now pull the rope with the same force at an angle to the horizontal (with the box remaining flat on the table), does the acceleration of the box (a) remain the same, (b) increase, or (c) decrease? Explain.

Since the box remains flat on the table the vertical acceleration of the box is zero. The horizontal acceleration of the box is proportional to the horizontal force on it. When you pull the box at an angle to the horizontal, the horizontal component of the force is less than when you pull the box horizontally with the same magnitude force.

An ice cube floats in a glass of water filled to the brim. What can you say about the density of ice? As the ice melts, will the water overflow? Explain.

Since the ice cube floats, the density of ice is less than the density of water. The mass of the ice displaces a volume of water which has the same mass as the ice. The mass of the ice doesn't change as the ice melts, so the volume displaced remains the same whether its is solid or liquid. So the level of the water in the glass remains the same as the ice melts and the water doesn't overflow.

A recipe for souffle specifies that the measured ingredients must be exact or the duffle will not rise. The reipe calls for 6 large eggs. The size of large eggs can vary by 10% according to the USDA. What does this tell you about how exactly you need to measure the other ingredients.

Since the size of the eggs can vary by 10% the uncertainty of the size of a large egg is +5 or-5. So the other ingredients only need to be measured within this range.

A heavy crate rests on the bed of a flatbed truck. When the truck accelerates, the crate remains where it is on the truck, so it, too, accelerates. What force causes the crate to accelerate?

Static friction between the bed of the truck and the crate, in the direction of the acceleration. It is static friction, not kinetic friction because the crate is not moving relative to the truck bed.

Two identical arrows, one with twice the speed of the other, are fired into a bale of hay. Assuming the hay exerts a constant frictional force on the arrows, the faster arrow will penetrate how much farther than the slower arrow? Explain

The faster arrow will penetrate four times farther than the slower arrow. Since the faster arrow has twice the speed it has four times the kinetic energy and the hay must do four times as much work to stop it. Assuming the frictional force is constant, this means it must travel four times as far.

Two spheres look identical and have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine which is which.

The hollow sphere will have a larger moment of inertia than the solid sphere since all its mass is far from the axis of rotation. So any experiment that involves the spheres rotating will be able to distinguish them. For example, roll the spheres down a rough incline, starting together from the same height. The solid sphere will reach the bottom first.

A light object and a heavy object have the same kinetic energy. Which has the greater momentum? Explain.

The in order to have the same kinetic energy the light object must be traveling faster than the heavy object, so it might seem that the light object would have the greater momentum. However, if we look at the relationship between momentum and kinetic energy we find KE = 12 mv2 = (mv)2 = p2 → p = 2mKE so the heavier object, which has the greater mass, also 2m 2m has the greater momentum.

Two vectors have length V1 = 3.5 km and V2 = 4.0 km. What are the maximum and minimum magnitudes of their vector sum?

The maximum magnitude is when the two vectors are parallel. In that case the magnitude of their sum is the sum of their magnitudes: 3.5 km + 4.0 km = 7.5 km. The minimum magnitude is when the two vectors are antiparallel. In that case the magnitude of their sum is the difference of their magnitudes: 4.0 km - 3.5 km = 0.5 km.

A bicycle odometer (which measures distance traveled) is attached near the wheel hub and is designed for 27 inch wheels. What happens if you use it on a bicycle with 24 inch wheels

The odometer counts revolutions and uses the radius of the wheel to calculate the distance traveled in each revolution; each revolution would be 27π inches. With 24 inch wheels the distance for each revolution is less than with 27 inch wheels (only 24π inches) so the odometer reading will be high (i.e., the distance the odometer reads will more less than the actual distance traveled).

Why is the CM of a 1 m length of pipe at its mid-point, whereas this is not true for your arm or leg?

The pipe is uniform so its center of mass is at its geometric center, i.e. its midpoint. Your arm or leg is not uniform so the CM is not at the midpoint.

Why is it more difficult to do a sit-up with your hands behind your head than when your arms are stretched out in front of you?

To do a sit-u your must rotate your upper body about an axis through your hips. With your arms behind your head your moment of inertia is greater than with your arms stretched out in front of you so it takes a larger torque to rotate your upper body.

Place yourself facing the edge of an open door. Position your feet astride the door with your nose and abdomen touching the door's edge. Try to rise o your tiptoes. Why can't this be done?

When you rise on your tiptoes your center of mass shifts forwards. But with your nose and abdomen against the door your CM can't shift forward and gravity exerts a torque on you which returns your feet to the floor.

A stone hangs by a fine thread from the ceiling, and a section of the same thread dangles from the bottom of the stone. If a person gives a sharp pull on the dangling thread, where is the thread likely to break: below the stone or above it? What if the person gives a slow and steady pull?

With a sharp pull the bottom thread is likely to break because of the stone's inertia resisting a change in it motion. With a slow and steady pull the top thread is more likely to break because the stone's weight acts on it in addition to the pull.

Does an apple exert a gravitational force on the Earth? If so, how large a force? Consider an apple (a) attached to a tree, and (b) falling.

Yes, an apple exerts a gravitational force on the Earth equal to its weight (by Newton's 3rd law). It is the same whether the apple is attached to a tree or falling.

Can the normal force on an object ever do work? Explain.

Yes, if the normal force has a component parallel to the displacement. For example, when you lift an object vertically with your hand, the normal force of your hand on the object is parallel to the displacement so it does work.

Can an object be increasing in speed as its acceleration decreases? If so, give an example. If not, explain.

Yes, if the rate at which the speed increases slows down. For example, consider a car that takes 10 seconds to accelerate from rest to 40 km/h but then takes 20 seconds to continue to accelerate to 80 km/h. The average acceleration during the 20 seconds is half what it is during the first 10 seconds.

Consider what happens when you push both a pin and the blunt end of a pen against your skin with the same force. Decide what determines whether your skin is cut—the net force applied to it or the pressure.

You can push the blunt end of a pen very hard against your skin without it penetrating while a much smaller force will cause the pin to penetrate your skin. Since the pin has a much smaller point than the pen, for the same force the pressure of the pin is much greater. It is pressure, not net force which determines whether your skin is cut.

A person sitting in an enclosed train car, moving at constant velocity, throws a ball straight up into the air in her reference frame. (a) Where does the ball land? What is your answer if the car (b) accelerates, (c) decelerates, (d) rounds a curve, (e) moves with constant velocity but is open to the air?

a) In her hands. (b) Behind her. (c) In front of her. (d) To her side that is on the outside of the curve. (e) Behind her because the ball is slowed by air resistance.

A box rests on the (frictionless) bed of a truck. The truck driver starts the truck and accelerates forward. The box immediately starts to slide toward the rear of the truck bed. Discuss the motion of the box, in terms of Newton's laws, as seen (a) by Mary standing on the ground beside the truck, and (b) by Chris who is riding on the truck.

a) Mary sees the box remain at rest, in accord with Newton's 1st law, and the truck move out from under it. (b) Chris sees the box that is initially at rest start to move without a net external force being applied to, violating Newton's 1st law because the truck is accelerating and therefore not an inertial reference frame.

Analyze the motion of a simple swinging pendulum in terms of energy, (a) ignoring friction, and (b) taking friction into account. Explain why a grandfather clock has to be wound up.

a) Start with the pendulum at the top of its swing where it stops for an instant and its energy is all gravitational potential energy. As it swings down potential energy is changed to kinetic energy but the total amount of mechanical energy remains constant. At the bottom of its swing it has it minimum amount of potential energy and its maximum amount of kinetic energy and therefore its maximum speed. At it swings up to the other side, kinetic energy is changed back to gravitational potential energy and at the top of its swing on the other side it has only potential energy and is at the same height as it was originally. It then swings down the other direction and the process repeats indefinitely.b) With friction (including air resistance) mechanical energy is changed to thermal energy so the total amount of mechanical energy does not remain constant. So on each swing the pendulum has less gravitational potential energy at the top of its swing and so the height of each swing is less. Eventually all the mechanical energy will be changed to thermal energy and the pendulum stops swinging. This is why a grandfather clock needs to be wound up: energy from the spring is used to replace the mechanical energy lost due to friction.

When a golf ball is dropped to the pavement, it bounces back up. (a) Is a force needed to make it bounce back up? (b) If so, what exerts the force?

a) Yes. The direction of the ball changes so it accelerates and there must be a force that causes the acceleration. (b) The pavement.

A projectile is launched at an angle of 30° to the horizontal with a speed of 30 m/s. How does the horizontal component of its velocity 1.0 s after launch compare with its horizontal component of velocity 2.0 s after launch?

f air resistance can be neglected the horizontal component of acceleration is zero and the horizontal component of velocity is constant so it would be the same 1.0 s after launch and 2.0 s after launch. If air resistance cannot be neglected there is a horizontal component of acceleration that causes the projectile to decelerate so the horizontal component of velocity at 1.0 s after launch would be greater than at 2.0 s after lauch.

A baseball player hits a foul ball straight up into the air. It leaves the bat with a speed of 120 km/h. In the absence of air resistance, how fast will the ball be traveling when the catcher catches it?

120 km/h. By symmetry, the speed of the ball at any point on its downward path is the same as its speed at the same point on its upward path. (The velocity is opposite.)

If the acceleration of an object is zero, are no forces acting on it? Explain.

Not necessarily. The necessary condition for the acceleration to be zero is that there is no net external force acting on the object. As long as the forces cancel out then the acceleration will be zero.

What happens to the gravitational potential energy when water at the top of a waterfall falls to the pool below?

On the way down gravitational potential energy is changed to kinetic energy of the water and to thermal energy of the water and the surroundings because of air resistance. When the water hits the pool the kinetic energy is changed to thermal energy, sound, and wave energy.

A person exerts an upward force of 40 N to hold a hag of groceries. Describe the "reaction" force (Newton's third law) by stating (a) its magnitude, (b) its direction, (c) on what object it is exerted and (d) by what object it is exerted.

The "reaction" force is (a) 40 N (b) downward (c) exerted on the person (d) by the bag of groceries.

The Sun's gravitational pull on the Earth is much larger than the Moon's. Yet the Moon is mainly responsible for the tides. Explain. [Hint: Consider the difference in gravitational pull from one side of the Earth to the other.]

Because the Moon is much closer to Earth than the Sun, the difference in the Moon's gravitational pull from the near side to the far side of the Earth is greater than for the Sun. It is this differential gravitational pull that causes the tides.

Why does a child in a wagon seem to fall backward when you give the wagon a sharp pull forward?

Because the child has inertia she tends to stay at rest unless a net external force acts on her (Newton's 1st law). The force is applied to the wagon, not the child, so it accelerates out from under her and the she falls backwards relative to the wagon.

Why can a bucket of water be whirled in a vertical circle without the water spilling out, eve3n at the top of the circle when the bucket is upside down.

Both the water and the bucket are falling around the circle with the same acceleration so there is no relative motion between them and the water stays in the bucket.

Why might your foot hurt if you kick a heavy desk or a wall?

By Newton's 3rd law, if your foot exerts a force on a desk or wall then the desk or wall exerts an equal but opposite force on your foot. It is that force that makes your foot hurt.

When you stand still on the ground, how large a force does the ground exert on you? Why doesn't this force make you rise up into the air?

By Newton's second law the force the ground exerts on you is equal to your weight. This doesn't make you rise up into the air because it is cancelled by your weight in the opposite direction and the net force on you is zero.

Why do you tend to lean backward when carrying a heavy load in your arms?

By leaning backward you keep the center of gravity of the system of you and the load over your base (i.e. your feet) which makes you more stabl

Why do bicycle riders lean inward when rounding a curve at high speed?

By leaning inward the normal force of the road on the bicycle has a component in the centripetal direction which adds to the static friction, also in the centripetal direction. The centripetal force is greater so the centripetal acceleration can be greater and the speed rounding the curve can be greater.

Seasoned hikers prefer to step over a fallen log in their path rather than stepping on top and jumping down on the other side. Explain.

By stepping over the log you don't have to raise your center of gravity as much as if you step on the log and therefore have to do less work against gravity.

The force of gravity on a 2 kg rock is twice as great as that on a 1 kg rock. Why then doesn't the heavier rock fall faster?

The 2 kg rock also has twice as much inertia so it takes twice as much force to have the same change in motion as the 1 kg rock

As a freely falling object speeds up, what is happening to its acceleration due to gravity— does it increase, decrease, or stay the same?

The acceleration due to gravity stays the same (unless the distance the object falls is very large, in which case it is not a good assumption to neglect air resistance and treat the object as freely falling.)

It is said that in ancient times a rich man with a bag of gold coins froze to death while stranded on a frozen lake. Because the ice was frictionless, he could not push himself to shore What could he have done to save himself had he not been so miserly?

He could have throw the bag of coins as hard as possible away from the shore. Because the initial momentum of the man and bag was zero, the final momentum must also be zero since. The man would have recoiled in the opposite direction that he threw the bag and slid to shore.

A pendulum is launched from a point that is a height h above its lowest point in two different ways. During both launches, the pendulum is given an initial speed of 3.0 m/s. On the first launch, the initial velocity of the pendulum is directed upward along the trajectory, and on the second launch it is directed downward along the trajectory. Which launch will cause it to swing the largest angle from the equilibrium position? Explain.

If air resistance and friction can be ignored the largest angle will be the same for both launches. The initial kinetic energy and the initial potential energy are the same so the total mechanical energy is the same. The largest angle occurs when the pendulum bob stops for an instant and the mechanical energy is completely gravitational potential energy. Since the total mechanical energy is the same the maximum gravitational potential energy is the same and the maximum angle is the same.

If only an external force can change the momentum of the center of mass of an object, how can the internal force of an engine accelerate a car?

It can't, at least not directly. Through the transmission the force of the engine is transferred to the tires. The tires exert a frictional force on the road so the road exerts an equal but opposite frictional force on the tires. It is this external force that accelerates the car.

What is wrong with the road sign? Memphis 7 mi (11.263 km)

Miles is rounded to 1 figure so km needs to be as well. Sig Figs.

We claim that momentum is conserved, yet most moving objects eventually slow down and stop. Explain

Momentum is only conserved if the net external force acting on an object or system of objects is zero. This is not usually the case for a moving object since friction, air resistance, and/or other resistive forces are acting on the object.

In archery, should the arrow be aimed directly at the target? How should your angle of aim depend on the distance to the target?

No, the arrow should be aimed slightly above the target to account for gravity pulling the arrow down as it travels to the target. Assuming you fire the arrow with the same initial velocity, the angle above the target should increase as the distance to the target increases.

If one object has a greater speed than a second object, does the first necessarily have a greater acceleration? Explain, using examples.

No. Acceleration is the rate at which the velocity is changing. For example, a car with a constant speed of 100 mph has zero acceleration while a bicycle whose speed changes from 5 mph to 10 mph in 5 minutes has a nonzero and therefore greater acceleration even though its speed is much less.

Can an object have a varying speed if its velocity is constant?

No. If the velocity is consent both the speed and direction are constant. (However, an object can have varying velocity if its speed is constant; for example, a car going around a corner at constant speed.)

When a "superball" is dropped, can it rebound to a height greater than its original height?

No. Since the ball starts from rest it has only gravitational kinetic energy initially. The potential energy is changed to kinetic energy on its way down and the ball has only kinetic energy just before hitting the ground, assuming that the ground is the reference level for gravitational potential energy. If no mechanical energy is lost during contact with the ground (i.e. if the ball is perfectly elastic) then the kinetic energy just after leaving the ground is the same and the ball will rebound to the same height it was dropped from, assuming air resistance is negligible. For real balls there is internal friction during the collision with the ground, kinetic energy is lost as thermal energy, and the ball rebounds to a height less than its original height.

In drag racing, is it possible for the car with the greatest speed crossing the finish line to lose the race? Explain.

Not for constant acceleration; in that case the car with the greatest acceleration would have the greatest speed crossing the finish line and the shortest time, so it would finish first. However, if the acceleration were not constant, then it would be possible. For example, compare a car with a large initial acceleration followed by constant velocity to a car with constant acceleration. The car with constant acceleration would have a higher speed crossing the line but would cross the line after the other car. See the diagram below, noting that the areas under the two curves must equal the length of the track and so be the same.

For an answer to be complete the units need to be specified. Why?

Not knowing what system it is completely changes the answer to the question,

If the net force on a system is zero, is the net torque also zero? If the net torque on a system is zero, is the net force zero?

Not necessarily in either case. For example in a couple (top diagram) the net force is zero but the net torque is not zero. The object will rotate counterclockwise without any translational motion. Similarly, in the bottom diagram, the net torque is zero but the net force is not zero. The object will move downward without rotating.

A girl is whirling a ball on a string around her head in a horizontal plane. She wants to let go at precisely the right time so that the ball will hit a target on the other side of the yard. When should she let go of the string?

She should let go of the string when the ball is at a position where the tangent to the circle points toward the target.

Suppose you lift as suitcase from the floor to a table. The work you do on the suitcase depends on which of the following: (a) Whether you lift it straight up or along a more complicated path, (b) the time it takes, (c) the height of the table, and (d) the weight of the suitcase.

Since muscular forces are nonconservative the work you do depends on all of the above except the time it takes.

Describe the energy transformations when a child hops around on a pogo stick

Start with the child at the top of her jump where she has her maximum amount of gravitational potential energy and zero kinetic energy. As she falls gravitational potential energy is changed to kinetic energy until the pogo stick makes contact with the ground. Then kinetic energy and some more gravitational potential energy are converted to elastic potential energy as the spring of the pogo stick compresses. At its maximum compression the system has its minimum amount of gravitational potential energy, zero kinetic energy, and its maximum amount of elastic potential energy. The elastic potential energy is changed back to gravitational potential energy and kinetic energy until the top of the next jump when it is all gravitational potential energy again. In the case of an ideal pogo stick (no internal friction) and no air resistance the height of the second jump will be the same as that of the first jump. For a real pogo stick and including air resistance, some of the mechanical energy is changed to thermal energy of the system and surroundings. In order to jump as high, the child must do work on the pogo stick and transfer some of her stored energy to the pogo stick

A block is given a push so that it slides up a ramp. After the block reaches its highest point, it slides back down but the magnitude of its acceleration is less on the descent than on the ascent. Why?

The acceleration is proportional to the net force parallel to and down the ramp. During the block's ascent, both the force of kinetic friction and the parallel component of the block's weight are down the ramp so their magnitudes add. During the block's descent the force of kinetic friction is up the ramp and the parallel component of the weight is down the ramp, so their magnitudes subtract and the net force down the ramp, and therefore the acceleration of the block, is less than during the ascent

Will the acceleration of a car be the same when the car travels around a sharp curve at a constant 60 km/h as when it travels around a gentle curve at the same speed? Explain.

The acceleration of the car will be greater for the sharp curve because it has a smaller radius of curvature and therefore the direction is changing more quickly.

Which one of these motions is not at constant acceleration: a rock falling from a cliff, an elevator moving from the second floor to the fifth floor making stops along the way, a dish resting on a table.

The acceleration of the elevator is not constant. Strictly speaking, the others are not at constant acceleration. However, if air resistance is negligible than the falling rock is nearly at constant acceleration. The dish on the table is moving along with the Earth's motion and so is accelerating. The Earth's motion is not at constant acceleration. However, in the reference frame for which the dish is at rest its acceleration is zero and is constant.

Why is it incorrect to think the more digits you represent in your answer, the more accurate it is?

The accuracy of an answer is determined by the accuracy of the physical measurements it is based on. Having more digits in the answer doesn't mean that the measurements were better. A calculation cannot add accuracy to an answer.

According to Eq. 6-5, the longer the impact time of an impulse, the smaller the force can be for the same momentum change, and hence the smaller the deformation of the object on which the force acts. On this basis, explain the value of air bags, which are intended to inflate during an automobile collision and reduce the possibility of fracture or death.

The airbag increases the impact time over which the stopping force acts on the occupants of the vehicle. This decreases the force acting on the person which reduces the possibility of injury or death.

Water balloons are tossed from the roof of a building, all with the same speed but with different launch angles. Which one has the highest speed on impact? Ignore air resistance

The all have the same speed on impact. They all have the same initial gravitational potential energy since they start from the same height and the same initial kinetic energy since they start with the same speed. Since we are ignoring air resistance they all have the same kinetic energy and therefore the same speed on impact. Note however that their final velocities are different since they impact at different angles.

At a hydroelectric power plant, water is directed at high speed against turbine blades on an axle that turns an electric generator. For maximum power generation, should the turbine blades be designed so that the water is brought to a dead stop, or so that the water rebounds?

The blades should be designed so that the water rebounds so that the change in momentum of the water is greater and the impulse on the blades is greater.

A rocket following a parabolic path through the air suddenly explodes into many pieces. What can you say about the motion of this system of pieces?

The center of mass of the system of pieces continues to follow the original parabolic path. Without more information it is impossible to say what happens with the individual pieces.

A squash ball hits a wall at a 45° angle as shown in the figure. What is the direction (a) of the change in momentum of the ball, (b) of the force on the wall?

The change in momentum is to the left. (b) The force of the wall on the ball is also to the left so, by Newton's 3rd law, the force of the ball on the wall is to the right.

Can the displacement vector for a particle moving in two dimensions ever be longer than the length of path traveled by the particle over the same time interval? Can it ever be less? Discuss.

The displacement vector can be equal to or less than the length of the path traveled by the particle, but it can never be longer. If the particle moves in a straight line without changing direction then the magnitude of the displacement vector will equal the length of the path traveled. In any other situation the displacement vector will be shorter than the length traveled.

Compare the effort (or force) needed to lift a 10 kg object when you are on the Moon with the force needed to lift it on Earth. Compare the force needed to throw a 2 kg object horizontally with a given speed on the Moon and on Earth.

The force needed to lift the object must be at least equal to the object's weight. Since the force of gravity on the Moon is about 1/6 that on Earth, the needed force is also about 1/6. To throw the object horizontally you must accelerate it. The force needed is given by Newton's second law and depends only on the acceleration and the object's mass. Since the mass is the same on the Moon as on Earth, the force is also the same.

A uniform meter stick supported at the 25 cm mark is in equilibrium when a 1 kg rock is suspended at the 0 cm end (as shown in the figure). Is the mass of the meter stick greater than, equal to, or less than the mass of the rock? Explain your reasoning.

The mass of the meter stick is equal to the mass of the rock. Since the meter stick is uniform its center of gravity is at its geometric center, i.e. the 50 cm mark. The lever arm for the rock is 25 cm and the lever arm for the weight of the meter stick is also 25 cm. Since the meter stick is in equilibrium the net torque must be zero, and since the lever arms are the same, the forces must be the same magnitude.

Suppose a car moves at constant speed along a hilly road. Where does the car exert the greatest and least forces on the road: (a) at the top of a hill, (b) at a dip between two hills, (c) on a level stretch near the bottom of a hill?

The net force on the car equals the difference between the normal force of the road on the car and the weight of the car. It must be in the centripetal direction. The greatest force exerted by the car on the road is at a dip between two hills where the centripetal acceleration is up so the normal force must be greater than the weight. The least force is at the top of a hill where the centripetal acceleration is down sot he normal force must be less than the weight. On the level road there is no centripetal acceleration so the normal force equals the weight.

Why can a batter hit a pitched baseball further than a ball tossed in the air by the batter?

The pitched baseball has a greater change in momentum since its horizontal component of velocity changes direction which the tossed baseball has an initial horizontal velocity of zero. Therefore the impulse is greater for the pitched baseball. Assuming the batter can swing the bat with equal force in either case, for the pitched baseball the time of collision must be greater so the final horizontal velocity of the pitched baseball is greater than for the tossed baseball. Since the horizontal velocity is greater the distance the ball travels is farther.

Suppose a disk rotates at constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk's angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

The point always has radial acceleration. It is constant if the angular velocity is constant and changes if the angular velocity changes. The point only has nonzero tangential acceleration if the angular velocity is changing. If it is changing uniformly, then the tangential acceleration is constant.

A sphere and a cylinder have the same radius and the same mass. They start from rest at the top of a n incline. Which reaches the bottom first? Which has the greater speed at the bottom? which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

The sphere reaches the bottom first and has the greatest speed at the bottom since it has a smaller moment of inertia than the cylinder and therefore has less rotational KE and more translational KE. Both objects have the same amount of gravitational potential energy at the top of the incline since they have the same mass, so they have the same total kinetic energy at the bottom of the incline. But since the cylinder is moving slower it has less translational KE and more rotational KE than the sphere.

according to Newton's third law, each team in a tug of war pulls with equal force on the other team. What, then, determines which team will win?

The team that wins is the team that exerts the largest force on the ground. By Newton's third law the ground exerts a larger force on that team than on the other team so there is an unbalanced force on the system of the two teams and the system accelerates in that direction, pulling the losing team across the line.

One car travels due east at 40 km/h, and a second car travels north at 40 km/h. Are their velocities equal? Explain.

Their speeds are equal but their velocities are not equal since they are traveling in different directions.

Compare the acceleration of a motorcycle that accelerates from 80 km/h to 90 km/h with the acceleration of a bicycle that accelerates from rest to 10 km/h in the same time.

They have the same acceleration since the change in velocity is the same for both.

Which pulls harder gravitationally, the Earth on the Moon, or the Moon on the Earth? Which accelerates more?

They pull with the same magnitude but opposite directions according to Newton's 3rd law. Because the Moon is less massive than the Earth, the Moon accelerates more.

Can two vectors of unequal magnitude add up to give the zero vector? Can three unequal vectors? Under what conditions?

Two vectors of unequal magnitude can never add to give the zero vector. Three unequal vectors can as long as the third vector goes from the tip of the second vector to the tail of the third vector. I.e., the three vectors form a closed triangle.

A small amount of water is boiled in a 1 gallon metal can. The can is removed from the heat and the lid put on. Shortly thereafter the can collapses. Explain.

When the water boils the can fill up with steam. After the can is removed from the heat and the lid put on the can cools and the steam condenses back to liquid water, leaving a partial vacuum in the can. Atmospheric pressure crushes the can since the force on the outside of the can is now much greater than the force on the inside.

Why do tightrope walkers carry a long, narrow beam?

With the beam, the moment of inertia of the system is greater than that of the tightrope walker alone. If the walker gets off center, gravity will exert a torque on the walker. With the beam the angular acceleration will be smaller and it will be easier for the walker to get centered again and keep from falling.

Mammals that depend on being able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body. On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

With the mass concentrated close to the body the legs have a smaller moment of inertia than if the mass were uniformly distributed. Thus less torque will be required to have a given angular acceleration, or, alternatively, a higher angular acceleration can be developed for the same torque. Thus the animal can run fast.

A coil spring of mass m rests upright on a table. If you compress the spring by pressing down with your hand and then release it, can the spring leave the table? Explain, using the law of conservation of energy.

Yes, if the elastic potential energy of the compressed spring is greater than the gravitational potential energy of the center of mass of the uncompressed spring. In that case when the spring becomes uncompressed again after releasing it, it has more energy than when it was initially uncompressed. The excess energy is kinetic energy which gets converted to gravitational potential energy as the spring leaves the table. Where did the excess energy come from, since energy can't be created or destroyed? It was transferred to the spring by your hand doing work on the spring, and came from energy you had stored in your body.

Is it possible for an object to receive a larger impulse from a small force than from a large force? Explain?

Yes, if the small force acts long enough since impulse is the product of the average force and the duration of the force.

Can an object have zero velocity and nonzero acceleration at the same time? Give examples.

Yes. For example, consider a car starting from rest. Its initial velocity is zero but its initial acceleration cannot be zero because if it were the velocity would remain zero and the car would not start moving. Another example is an object thrown straight upwards when it is at its highest point. The object stops for an instant so its velocity is zero but its acceleration is the acceleration due to gravity and so is nonzero.

Can an object have a northward velocity and s southward acceleration. Explain.

Yes. If the velocity and acceleration are in opposite direction then the object slows down.

Can a small force ever exert a greater torque than a larger force? Explain.

Yes. Since torque is force times lever arm, a small force with a large enough lever arm can exert a greater torque than a larger force with a smaller lever arm.

You measure the radius of a wheel to be 4.16cm. If you multiply by 2 to get the diameter, should you write the result as 8cm or 8.32cm?

You should record the answer as 8.32.

Two cannonballs, A an dB, are fired from the ground with identical initial speeds, but with θA larger than θB. (a) Which cannonball reaches a higher elevation? (b) Which stays longer in the air? (c) Which travels farther?

a) Cannonball A reaches a higher elevation because it has a greater initial vertical velocity. (b) Cannonball A stays in the air longer for the same reason. (c) The cannonball with a launch angle closest to 45° will travel farther because the range is greatest for a launch angle of 45°.

Can the magnitude of a vector ever (a) be equal to one of its components, or (b) be less than one of its components?

a) If a vector is parallel to a coordinate axis it only has one nonzero component and its magnitude is equal to the magnitude of that component. (b) The magnitude of a vector can never be less than any of its components.

A Superball is dropped from a height h onto a hard steel plate (fixed to the Earth), from which it rebounds at very nearly its original speed. (a) Is the momentum of the ball conserved during any part of this process? If we consider the ball and Earth as our system, during what parts of the process is momentum conserved? (c) Answer part (b) for a piece of putty that falls and sticks to the steel plate.

a) The momentum of the ball is not conserved during any part of the process since external forces act on it (gravity as it falls and rebounds, the force of the plate during the collision). (b) For the Earth-ball system all the forces are internal forces so the momentum of the system is conserved during all parts of the process. (c) For the Earth-putty system all the forces are internal forces so the momentum of the system is conserved during all parts of the process.

A ground retaining wall is shown in part (a) of the figure. The ground, particularly when wet, can exert a significant force F on the wall. (a) What force produces the torque to keep the wall upright? (b) Explain why the retaining wall in part (b) of the figure would be much less likely to overturn than that in part (a).

a) The weight of the wall exerts the torque to keep it upright. (b) The lever arm for the wall in (a) is small (half the width of the wall) so the torque due to its weight is small. For the wall in (b), in addition to the weight of the wall there is a torque due to the weight of horizontal part of the wall and the soil above it. This is a much larger force and has a much larger lever arm so the horizontal force exerted by the ground on the vertical part of the wall would have to be many tines larger in order to overturn it,

An object that is thrown vertically upward will return to its original position with the same speed as it had initially if air resistance is negligible. If air resistance is appreciable, will this result be altered, and if so, how? [Hint: The acceleration due to air resistance is always in a direction opposite to the motion.]

f air resistance is appreciable, the speed when the object returns to its initial position is less than its initial speed. The air resistance breaks the symmetry—on the way up air resistance is down and on the way down air resistance is up, unlike gravity which is always down.