Chapter 4

Questions:

1) Yes. An object will keep moving in straight-line motion unless acted upon by an external force. This is Newton's 1st Law.

2) On Earth, we can do very well and reduce friction (including air resistance) in a tube and see an object travel with constant velocity. Unfortunately we are limited in how long the tube can be. In space this can be directly tested and has been tested many times. The voyager space probe has left our solar system on its trip through deep space.

4) The child is at rest and wants to remain at rest (Newton's 1st Law - Law of Inertia). When the wagon is pulled, the part of the body that is not touching the wagon will "fall back", really just try and remain in place.

7) Newton's 1st Law - Law of Inertia. The china and glassware want to remain at rest and by pulling quickly, the tablecloth can exert its friction on them for a very short time; not enough for them to move.

8) Your natural state of motion is a straight line(Newton's 1st Law - Law of Inertia). In order to move around a curve, you must change your direction of motion (i.e. velocity). This means you must accelerate. In order to accelerate around the curve, the door of the car (or seatbelt) pushes on you and causes you to accelerate.

9) The head wants to remain at rest. If you are hit from behind, the seat pushes on your back, but if you do not have a headrest, there is nothing pushing on your head. Your body will move forward, while your head remains (momentarily) in place. It appears to whip back, but in fact it is remaining still(Newton's 1st Law - Law of Inertia).

12) Stopping and starting (acceleration) take more force (energy) than just maintaining a constant speed(Newton's 2nd Law - F = ma). This force is supplied by the engine which burns gasoline to work.

18) When you walk on a floating log, you are pushing backward on the log to propel yourself forward. In fact it is the log pushing back on you that propels you forward. Although the log is much more massive, it is still small enough that your force pushing backwards will cause it to move.

Exercises:

1)

3)

5) The force pulling forward is , the force pulling back is which makes the net force , so

6) , so

10) My weight = 190lb, converting to mass, m (kg), . My "weight" is the force of gravity acting on me, . Remember that the negative sign is simply telling me the direction of the force, down.

Chapter 5

Questions:

1) On moon, , so if my mass m = 85kg, my weight is . On Earth, , so my weight is , nearly six times as big.

2) My weight in pounds is W = 185 lbs, converting to kg, . My mass on the moon will be exactly the same.

3) Force of gravity pulling down is the same magnitude as the force of the ground pushing up. The net force is 0N, so you don't fly up.

5) If we know the magnitude and direction of the net Force, then we can calculate the acceleration from Newton's 2nd Law.

10) The speed is slowest at the top of the arc. The horizontal speed is always the same, but the vertical speed at the top of the arc is .

11) You should remain where you are. As the bullet approaches, it will fall. If you drop out of the tree then you will fall at the same rate as the bullet and you will be hit. By staying in the tree, the bullet falls and you will not be hit.

12) Vertical component of velocity is greatest at the beginning AND end of the arc.

13) In order to get the longest range on a broad jump, an athlete should try and jump as fast they can at an angle of about 45 degrees. If they jump more vertically, they will remain in the air longer, but their horizontal velocity will be smaller and will not carry them as far. If they jump more horizontally then they will have a smaller vertical velocity and will not remain in the air as long.

14) Air resistance slows things down. On the way up, the air resistance and gravity will slow the object down faster than with only gravity. The object will not reach as high a point. On the way back down, since it doesn't fall as far, and also the air resistance is slowing it down, the object will not move as quickly as the case without air resistance.

Exercises:

1) The astronaut has a mass of m = 70kg.

a)

b)

c)

d)

2)

The two forces must be added as vectors. From the drawing, the two forces act on the mass in perpendicular directions. The net force can be constructed by using the head-to-tail method. The Net force is found by using the Pythagorean Theorem.

9) The initial velocity is pure horizontal, so that . It takes for the diver to hit the water. This tells us that the cliff must have been a height of . To find the distance from the cliff the diver hits, we use .

10) The initial speed is pure horizontal which means that . We need to find how long it takes the tiger to fall the distance of 15m. we can then solve for and finally Now we can find the distance from the rock where the tiger lands,

Chapter 6

Questions:

1) If a bucket of water is swung in a vertical circle at a high enough speed, the water won't spill at the top of the circle when the bucket is upside down. Explain.

Answer: In order to move on a circular path, the NET force on the bucket (and the water) will be the centripetal force. The water has two forces exerted on it, the force of the bucket and the force of gravity. At the top of the circle, the centripetal force, which points toward the center of the circle the bucket is moving on, will be pointed straight down. The force of gravity on the water will also be straight down. If the speed of the bucket is too small, then the force of gravity will be larger than the centripetal force needed to stay moving on the circle, and the water will come out. If the speed is fast enough, then the centripetal force needed will be greater than the force of gravity, which means the bucket will push down on the water as well. In other words, when the centripetal force is greater than the force of gravity, the bucket (with water in it) is accelerating faster than gravity, so the water won't fall out.

3) Why does a car tend to skid on an icy curve?

Answer: If the road is icy, the force of friction (which keeps the car on the road) will be smaller and the car will want to keep moving in a straight line instead of turning. This means the car will skid off the road.

4) Explain why a car is less likely to skid when traveling around an icy level curve if it does so at a low speed?

Answer: Friction is the force that allows the car to make it around a turn. The tires of the car push on the road and the force of friction of the road pushes back on the car enough to keep the car moving around the circular path. When the road is icy, the strength of the friction force is reduced but not completely gone. If the car travels around the curve at slow speed, then it will not require much acceleration to make it around the curve, and the smaller force of friction should be able to supply it.

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

Answer: No the acceleration will be greater for the sharp curve than for the gentle curve. Because the formula for centripetal acceleration is , if the speed (v) doesn't change, then the bigger the r is, the smaller ac will be. A gentle curve has a bigger radius than a sharp curve.

9) 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.

Answer: Yes an apple exerts a force on the earth, because as Newton's 3rd law states, for every force, there is an equal but opposite reaction force. The earth pulls on the apple with a gravitational force, so the apple must pull with the same strength force, but pulls up on the earth.

a) If the apple is attached to a tree, it has two forces acting on it, the force of the earth (gravity) on it, and the force of the tree holding it up. The NET force is 0N. So the force of the apple on the earth is the same as the force of the earth on the apple, its weight.

b) When the apple is falling, the only force acting on it is the force of the earth which is the same as before, so the apple still pulls up on the earth with the same force as before, its weight.

12) Is the acceleration due to gravity the same everywhere?

Answer: No. It basically depends on how far you are from the center of the Earth, the further away, the weaker the force of gravity on you and also the weaker the acceleration of gravity. It can also be affected by differences in the density of the Earth. There is a technique for prospecting which involves measuring slight changes in the Earth's gravitational pull in order to find metal ore which will tend to pull a little stronger than rocky material which is less dense.

14) Would your weight on top of Mt. Everest be more or less than your weight at sea level?

Answer: Your weight would be smaller at the top of Mt. Everest because according to Newton’s law of gravitation, , the force is smaller the further two masses are from each other. In this case, one of the masses is you, the other is the earth, and the distance between them is the distance between their centers. Since the top of Mt Everest is further from the center of the Earth, your weight is smaller.

23) The sun's gravitational pull on the earth is much larger that the moon's. Yet the moon is mainly responsible for the tides. Explain.

Answer: Although the sun does have some impact on the earth's tides, the tidal effect is caused by differences in the gravitational force of attraction between the near-side and far-side of the earth. Because teh sun is much further from the earth than the moon, there is very little difference between how strongly the sun pulls on either side of the earth. The moon being much closer to the earth, is always exerting a stronger pull on the near-side of the earth than the far-side, causing the tides.

Exercises:

1) 1. A racing car travels around a curve of radius 30 meters at a speed of 30 m/s. What is the centripetal acceleration? Express the answer in g's; that is how many times larger that the acceleration of gravity, , is this?

Answer: The formula for centripetal acceleration is , so to convert to g's, .

2) What force is required in the above exercise if the car's mass is 1400kg?

Answer: From above, , so that the centripetal force is

3) If a stone is spun with a speed of 2.5m/s in a circle of radius 0.5m, what is its centripetal acceleration? If the stone's mass is 2kg, what force is being exerted on it and in what direction?

Answer: Using the formula for centripetal acceleration, and the centripetal force will be , pointed toward the center of the circle.

4) A jet plane traveling 300m/s pulls out of a dive in an arc of radius 2500m. What was the plane's acceleration in "g's" (i.e., what multiple of the acceleration of gravity, g)?

Answer: From the formula for centripetal acceleration, . This is the equivalent of

5) A child moves with a speed of 1.80m/s when 12 m from the center of the merry-go-round. Calculate (a) the centripetal acceleration of the child, and (b) the net force exerted on the child (mass = 25kg).

Answer: Using the formula for centripetal acceleration, and the centripetal force will be , pointed toward the center of the circle.

12) If the moon changed its orbit around the earth so that it was only half as distant as it is now, by what factor would the gravitational force on the moon due to the earth be changed?

Answer: If the distance between the earth and moon gets smaller, then the gravitational force will get stronger. The normal gravitational force is , and the new gravitational force will be , or 4 times as strong as it normally is.

13) Suppose both the mass of the earth and the mass of the moon were double their present values, but that the distance between them remained the same. By what factor would the force of gravity between them change? Would the speed of the moon around the earth have to increase, decrease, or remain the same?

Answer: If the mass of the moon and earth both double, then the gravitational force will go up. The normal force of gravity is , and the new force of gravity would be , or 4 times as big. Because the force of gravity is the reason that the moon remains on a circular orbit, we can understand that the gravitational force is the centripetal force acting on the moon which makes it move on a circular path. Without the centripetal force, the moon would take off on a straight-line path. The formula for centripetal force is . Because the distance between the earth and moon remains the same, if the moon is to remain on a circular orbit, then the gravitational force should match the centripetal force, which means that the centripetal force has to increase by a factor of 4. This means the velocity of the moon must increase (by a factor of 2).

Chapter 7

Questions:

1) Work depends on what two factors?

Answer: Work depends on force and distance. Work = Force x distance or W=Fd

3) In which case is more work done: when a 50 kg bag of groceries is lifted 50 cm or when a 50 kg crate is pushed 2 m across the floor with a force of 50 N?

Answer: When lifting a 50 kg bag of groceries, the force you will use is equal to the weight of the bag (you are working against the force of gravity). The force you must exert is . The work done to life the bag 50 cm = 0.50 m is:

To push the crate 2 m the work is: , so it takes more work to life the groceries.

6) A moving billiard ball strikes a second ball and imparts some kinetic energy to the latter. Did it do work on the second ball?

Answer: Yes. Because the some (or all) of the kinetic energy was transferred to the second ball, the first ball changed the second ball’s energy. The only way to do this is if the first ball did work on the second ball. The amount of work done on the second ball is equal to the kinetic energy the second ball now has.

8) Which has more kinetic energy, a 1000kg car traveling 80km/hr or a 2000kg car traveling 40km/hr?

Answer: We can compare the amount of energy in the 2 cars without really calculating the amount in each car. The first car has a kinetic energy of and we know that so that second car has a kinetic energy of , or half as much as the first car.

10) Does a stretched rubber band have potential energy? How do you know?

Answer: Yes it does. When you let a stretched rubber band go, it starts moving (has kinetic energy). Since energy is neither created nor destroyed it must have gotten the kinetic energy from the rubber band returning to its original length (un-stretching). This means the stretched rubber band had some potential energy (recall that potential energy is the potential for motion).

12) Describe all the energy transformations that take place when you throw a ball into the air, when the ball reaches its maximum height and descends, and finally, when you catch it.

Answer: Before you begin to throw the ball, the total energy of the ball is 0J. When you begin to throw the ball into the air, your muscles use chemical energy to do work on the ball, giving it kinetic energy. As it leaves your hand, it has the maximum velocity and the maximum kinetic energy. As the ball rises, it loses kinetic energy but gains gravitational potential energy. When the ball is at its highest point, it has as much gravitational potential energy as it had kinetic energy leaving your hand. This is because energy is conserved. As the ball begins to fall back down, it loses gravitational potential energy, but gains kinetic energy. When you catch the ball, your muscles again use chemical energy to do work on the ball, but in this case, they take energy away from the ball, so the the final total energy of the ball is zero again.

15) What is the purpose of a spring that must be wound up in a pendulum clock?

Answer: Because there is a small amount of friction, the pendulum loses a small amount of energy with each swing, and the pendulum would have a smaller and smaller amplitude until it finally stopped. The potential energy stored in the spring can restore the energy lost to friction so that the pendulum keeps swinging with the same amplitude.

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

Answer: No. If it did then it would have more gravitational potential energy than when it started, and also a more total energy than it started with. The only way to change energy is if something does work on it. For example, pushing with your hand as it falls would make it bounce to a higher height than when it started, because your hand does work on it (and increases the total energy).

Exercises:

1) 1. A horse exerts a 900 N force to pull a 600 kg load 2.5 km. How much work did the horse do?

Answer: Do not be confused by extra information. All you need to find the work done is, a) the force and b) the distance the force is exerted. In addition the force should be in Newtons (N) and the distance in meters (m). Use 2.5 km = 2500 m.

2) A 50 kg woman climbs a flight of stairs 6.0 m high. How much work is required?

Answer: We need the force and the distance. The force to lift the woman is the same as her weight. , so that the work is

3) A 550 kg crate rests on the floor. How much work is required to move it at a constant speed (a) 2.0 m along the floor against a frictional force of 150 N and (b) 2.0 m vertically.

Answer: (a) In order to move the crate at a constant speed, the NET force must be 0 N. This is because according to Newton’s 2nd law, the net force will case the mass to accelerate, F = ma. If the crate moves with a constant speed, then a = 0 m/s2 and F = 0 N. If there is a friction force of 150 N, then 150 N is necessary to work against the force of friction and keep the crate moving with constant speed.

To lift the crate, the force applied is equal to the weight of the crate or and the Work is then,

4) How far must a 200 kg pile driver fall if it is to be capable of doing 13000J of work?

Answer: Lifting the pile driver up gives it potential energy. When the pile driver falls, it loses potential energy and gains kinetic energy. It loses all it’s potential energy (converted to kinetic energy) by the time it hits the post, and rams the post into the ground. In the process of ramming the post, it comes to rest, losing all its kinetic energy. It lost its kinetic energy by doing work on the post. This means that the work done on the post comes from the kinetic energy which comes from the potential energy. If the original gravitational potential energy was 13000J, then form the formula for gravitational potential energy, , we can find h, .

5) Six bricks, each 6.0cm thick with mass 1.5kg lie flat on a table. How much work is required to stack them one on top of another?

Answer: The bottom brick doesn't need to be moved. When we stack the next brick on top of it, we are lifting the second brick by 6.0cm or 0.060m. The amount of work required to do this is just the amount of gravitational potential energy gained by the brick in doing this. If we say the bricks have no gravitational energy sitting on the table then the gain in gravitational energy is . In order to lift the third brick on top of the first two, we will lift it a total of 12.0cm or 0.120m, for a gain of gravitational potential energy of . The following table shows how the bricks are stacked

 Brick number height, h(m) 1 0 0 2 0.060 0.88 3 0.120 1.8 4 0.180 2.6 5 0.240 3.5 6 0.300 4.4
The total work done is

6) Calculate the kinetic energy of a 55kg person running 9.0m/s

Answer: The kinetic energy formula is

7) When a bicycle's speed is doubled, by what factor does its kinetic energy change?

Answer: The original kinetic energy will be . When the speed is doubled, the new kinetic energy will be or 4 times as large.

15) A car rolls from rest down a hill with no friction. At the bottom of the hill its speed is 10 m/s. How high is the hill?

Answer: At the top of the hill the car has only gravitational potential energy. At the bottom of the hill it has lost all of its gravitational potential energy but gained kinetic energy. From the Law of Conservation of Energy, the total energy of the car can not change, only change from one form to another; in this case, gravitational energy is turned into kinetic energy. The total energy of the car at the bottom of the hill is just it's kinetic energy, , but at the top of the hill its total energy is just its gravitational potential energy, . These energies have to be the same, because they are both the total energy of the car. Setting them equal to each other, we get . The mass cancels from both sides and , or , so that . Finally we can solve for h,

19) Suppose a car starts coasting from the top of a hill that is 50m high. How fast will it be going at the bottom of the hill if there is no friction?

Answer: As the above problem, , so that or and then

so that

Chapter 8

Questions:

2) Name three everyday experiences that illustrate the conservation of momentum principle.

Answer: Any time a collision takes place, there is conservation of momentum. For example while playing billiards (a.k.a. pool), each time you hit the white ball and it strikes another ball, there is conservation of momentum in the collisions.

If you area rollerblading (or skateboarding) with a friend and you push off each other from rest, your total momentum before you push off each other is 0, and will also be zero after you push off each other. This means that one person will have positive momentum and the other will have negative momentum.

When a bat hits a baseball, it slows down or even stops after contact, and the baseball goes flying into the air (taking the momentum the bat had).

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

Answer: It is easiest to make up some numbers to use. Let's use a mass of 1kg and a mass of 10kg, and a momentum of . Because the momentum is p = mv, then . The 1kg mass will have and the 10kg mass will have . Clearly the one with less mass will have to have a greater velocity in order to have the same momentum. The kinetic energy can also be found, and for the 1kg mass is , while the kinetic energy for the 10kg mass is . Because kinetic energy depends on mass times velocity squared, the velocity is more important at changing the kinetic energy than the mass is.

5) Is it possible for an object to have momentum without having energy? Can it have energy without having momentum?

Answer: Yes. If an object has momentum then it has both mass and velocity. If an object has velocity then it must have kinetic energy. No. An object can have potential energy and no kinetic energy. If it has no kinetic energy then it has no velocity and so it has no momentum.

6) Is it possible for a body to receive a larger impulse from a small force than from a large force?

Answer: Yes. Impulse is force times time. A small force acting over a long time can easily provide more impulse (change in momentum) than a larger force which acts for a much shorter time.

9) In a collision between two cars, which would you expect to be more damaging to the occupants: if the cars collide and remain together or if the two rebound backward? Explain.

Answer: If the two cars remain together the occupants of the car will be more damaged. When the cars remain together after teh collision, the collision is said to be inelastic. In that case, mechanical energy is not conserved, but goes into the twisting of metal, the sound of the metal crunching and the glass breaking, as well as heat energy. In an elastic collision (the cars rebound backward) energy is conserved as well as momentum. In other words, the kinetic energies before and after the collision are roughly the same, and very little energy will go into damaging the cars and their occupants.

12) It is said that in ancient times a rich man with a bag of gold coins was frozen to death stranded on the surface of 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?

Answer: He could have saved himself by forcefully throwing the gold coins in the opposite direction that he wanted to go. Because the man and coins together before being thrown have a total momentum of 0, after the coins are thrown, the total momentum must still be 0. If the coins are thrown in the negative direction, they will have negative momentum, which means the man must have positive momentum and can get to shore.

17) When you release a balloon that has just been inflated, why does it fly across the room?

Answer: The initial total momentum of the balloon and air before the air is released is 0. After the air is released, the final total momentum must still be zero, because of conservation of momentum. When the air is expelled by the balloon, it has negative momentum (assuming the air is blowing in the negative direction) which means that the balloon must have a positive momentum.

Exercises:

2) A 90-kg fullback is travelling 5.0 m/s and is stopped by a tackler in 1.0s. Calculate (a) the original momentum of the fullback, (b) the impulse imparted to the tackler, and (c) the average force exerted on the tackler.

Answer: a) the fullback's original (initial) momentum is

b) The impulse imparted to the tackler is the gain in momentum of the tackler, which is exactly the loss of momentum by the fullback. The fullback loses all the momentum he had in part a), which means that the tackler gets an impulse of .

c) Impulse is also found by (Force) x (time), so

3) A car traveling 30km/hr strikes an identical car from the rear and the two lock bumpers. Assuming that the brake of the second car is off and that momentum is conserved in the collision, what is the velocity of the two attached cars after the collision?

Answer: To solve this we assume that the second car was initially at rest. Then we can calculate the total initial momentum as . The final momentum when the two lock bumpers is and they must be equal. So setting them equal to each other gives us . dividing both sides by 2m gives us,

4) If you throw a 4-kg rock from your resting boat with a speed of 10m/s, what will be the resulting speed (and direction) of your boat? (Total mass including you is 110kg).

Answer: The momentum of the rock and boat before the rock is thrown is 0, because as the problem states, the boat is at rest. When the rock is thrown, it will now have momentum (let's say negative) and it is . The boat (and you on it) must have a momentum exactly opposite (sign) the momentum of the rock, since the sum of the boat and rock momentums is 0 from the conservation of momentum principle. The momentum of the boat is , and this is how we find the velocity of the boat,

6) A 120-kg tackler traveling 3m/s tackles a 75-kg halfback running 6m/s in the opposite direction. What is their common speed immediately after the collision?

Answer: Suppose the tackler has a postitive velocity, then the tackler's initial momentum is , but the halfback with a negative velocity (opposite direction) has a momentum of . Clearly the momentum of the tackler is greater, and so when the two collide, the resulting momentum will be positive, which means they will move together in the same direction as the tackler is initially moving. The initial total momentum is After the tackler tackles the fullback, they move together, so that the total final momentum is and we can solve for the final velocity as