X 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.

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.

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.

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.

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.

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.

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 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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 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.

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).

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.

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.

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.

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.

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.

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.

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.

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,