Physics Chapter 5

Astronauts who spend long periods in outer space could be adversely affected by weightlessness. One way to simulate gravity is to shape the spaceship like a cylindrical shell that rotates, with the astronauts walking on the inside surface (Fig. 5-32). Explain how this simulates gravity. Consider (a) how objects fall, (b) the force we feel on our feet, and (c) any other aspects of gravity you can think of.

(A) if a passenger held an object beside their waist and then released it, the object would move in a straight line, tangential to the circle in which the passenger's waist was moving when the object was released. In the figure, we see that the released object would hit the rotating shell, and so fall to the floor, but behind the person. The passenger might try to explain such motion by inventing some kind of "retarding" force on dropped objects, when really there is no such force. (B) the floor exerts a centripetal force on the feet, pushing them towards the center. This force has the same direction ("upwards", away from the floor) that a passenger would experience on earth, and so it seems to the passenger that gravity must be pulling them "down". Actually pm the passengers are pushing down the floor, because the floor is pushing up on them. (C) the "normal" way of playing catch, for example, would have to change. Since the artificial gravity is not uniform, passenger would have to relearn how to throw something across the room to each other. There would not be projectile Motion as we experience it on earth. Also if the cylinder were small, there might be a noticeable difference in the acceleration of our head vs our feet. Since the head is closer to the center of the circle than our feet, and both the head and feet have the same period of rotations the centripetal acceleration (aR= 4pi^2r/ T^2) is smaller for the head. This might cause dizziness or a light headed feeling.

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

A satellite remains in orbit due to the combination of gravitational force on the satellite directed towards the center of the orbit and the tangential speed of the satellite. First, the proper tangential speed had to be established by some other force than the gravitational force. Then, if the satellite has the proper combination of speed and radius such that the force required for circular motion is equal to the force of gravity on the satellite, then the satellite will maintain circular motion.

Why do airplanes bank when they turn? How would you compute the banking angle given its speed and radius of the turn?

Airplanes bank when they turn because in order to turn, there must be a force that will be exerted towards the center of a circle. By tilting the wings, the lift force on the wings has a non-vertical component which points toward the center of the curve, providing the centripetal force. The banking angle can be computed from the free - body diagram. The sum of vertical forces must be zero for the plane to execute a level turn, and so Fcos 0 =mg . The horizontal component of the lifting force must provide the centripetal force to move the airplane in a circle.

Will an object weigh more at the equator or at the poles? What two effects are at work? They oppose each other?

An object weighs more at the poles, due to two effects which complement (not oppose) each other. First of all, the earth is slightly flattened at the poles and expanded at the equator, relative to a perfect sphere. Thus the mass of the poles is slightly closer to the center, and so experiences a slightly larger gravitational force. Secondly, objects of the equator have a centripetal acceleration due to the rotation of the earth that objects at the poles do not have. To provide that centripetal exhilaration, the apparent weight (the radially outward normal force of the earth on an object) is slightly less than the gravitational pull inwards. So the two affects both make the weight of an object at the equator less than at the poles.

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

For the water to remain in the bucket, there must be a centripetal force forcing the water to move in a circle along with the bucket. That's centripetal force gets larger with the tangential velocity of the water, since Fr =mv^2/ r. The centripetal force at the top of the motion comes from a combination of the downward force of gravity and the downward normal force of the bucket on the water. If a bucket is moving faster than some minimum speed, the water will stay in the bucket. If the bucket is moving too slow, there is insufficient force to keep the water moving in a circular path, and it spills out.

Would it require less speed to launch a satellite a) toward the east or b) toward the west? Consider the Earths rotation direction.

In order to orbit, a satellite must reach an orbital speed relative to the center of the earth. Since the satellite is already moving eastward when launched (due to the rotation speed at the surface of the earth) it requires less additional speed to launch it east to obtain the final orbital speed.

If the Earth's mass were double what it is, in what ways would the Moons orbit be different?

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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 line to the circle at the balls location, when extended, passes through the targets position. That tangent line indicates the direction of the velocity at that instant, and if the centripetal force is removed, then the ball will follow that line horizontally. See the top view diagram.

When will your apparent weight be the greatest, as measured by a scale in a moving elevator: when the elevator (a) Accelerates downward (b) accelerates upward (c) is in free fall (d) moves upward at a constant speed? In which case would your weight be least? When would it be the same as when your on the ground?

The apparent weight (the normal force) would be largest when the elevator is accelerating upward. From the free body diagram, with up as positive we have Fn - mg =ma Fn =m (a +g). With a positive acceleration, the normal force is greater than your weight. The apprentice weight would be the least when in free fall, because the apparent weight is zero since a =-g. When the elevator is movin with constant speed, your apparent weight would be the same as it is on the ground, since a =0 and Fn =mg.

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

When they lean inward bike tires push down on the ground at an angle. The road surface pushes back on the tires both vertically to provide a normal force which counteracts gravity and horizontally toward the center of the circle to provide the centripetal force enabling them to turn.

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.

The apple does exert a gravitational force on the earth. By newtons third law, the force on the earth due to the Apple is the same magnitude as the force on the Apple due to the earth— the weight of the Apple. The force is also independent of the state of motion of the Apple. So for both the hanging apple and a falling apple, the force on the earth due to the Apple is equal to the weight of the Apple.

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 centripetal acceleration for an object moving in a circular motion is inversely proportional to the radius of the curve, given a constant speed (a=v^2/r). So for a gentle curve (which means a large radius), the acceleration is smaller, while for a sharp curve (which means a small radius), the acceleration is larger.

Is the centripetal acceleration of Mars in its orbit around the sun larger or smaller than the centripetal acceleration of the earth?

The centripetal acceleration of mars is smaller than that of earth. The acceleration of each planet can be found by diving the gravitational force on each planet by the planets mass. The resulting acceleration is inversely proportional to the square of the distance of the planet from the sun. Since mars is further from the sun than the earth is, the acceleration of mars will be smaller. Also see the equation below.

The Sun's gravitational pull on the Earth is much larger than the Moon's. Yet the Moon's is mainly responsible for the tides. Explain.

The difference in force on the two sides of the earth from the gravitational pull of either the sun or the moon is the primary cause of the tides. That difference in force comes from the fact that the two sides of the earth are a different distance away from the pulling body. Relative to the sun, the difference in distance (earth diameter) other two sides of the sun, relative to the average distance to the sun, is given by 2Rearth/ Rearth to sun = 8.5 x 10^-5. The corresponding relationship between the earth and the moon is 3.3 x 10 ^-2. Since the relative change in distance is much greater for the earth and moon combination, we see that the moon is the primary cause of the earths tides.

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 force that the car exerts on the road is newtons third law reaction to the normal force of the road on the car, and so we can answer this question in terms of the normal force. The car exerts the greatest force on the road at the dip between two hills. There the normal force from the road has to support the weight and provide a centripetal upward force to make the car move in an upward curved path. The car exerts the least force on the road at the top of a hill. We have all felt the "floating upwards" and station as we have driven over the crest of a hill. In that case, there must be a net downward centripetal force to cause the circular motion, and so the normal force from the road does not completely support the weight.

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

The gravitational pull is the same in each case, by newtons third law. The magnitude of the pull is given by F=G Mmoon Mearth/ r^2. To find the acceleration of each body, the gravitational pulling force is divided by the mass of the body. Since the moon has the smaller mass, it will have the larger acceleration.

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 Earth?

The moon is not pulled away from the earth because both the moon and the earth are experiencing the same radial acceleration due to the sun. They both have the same period around the sun because they are both, on average, the same distance form the sun, and so they travel around the sun together.

Sometimes people say that water is removed from clothes in a spin dryer by centrifugal force throwing the water outward. What is wrong with this statement?

The problem with the statement is that there is nothing to cause an outward force, and so the water removed from the clothes is not thrown outward. Rather, the spinning drum pushes INWARD on the clothes and water. But where there are holes in the drum, the drum can't push on the water, and so the water is not pushed in. Instead, the water moves tangentially to the rotation, out the holes, in a straight line, and so the water is separated from the clothes.

How many "accelerators" do you have in your car? There are at least three controls that can be used to cause the car to accelerate. What are they? What accelerations do they produce?

The three major "accelerators" are the accelerator pedal, the brake pedal, and the steering wheel. The accelerator pedal (or gas pedal) can be used to increase speed (by depressing the pedal) or to decrease speed in combination with friction (by releasing the pedal). The brake pedal can be used to decrease speed by depressing it. The steering wheel is used to change direction, which also is an acceleration. There are other controls which could be considered accelerators. The parking brake can be used to decrease speed by depressing it. The gearshift lever can be used to decrease speed by downshifting. If the car has a manual transmission, then the clutch can be used to decrease speed by depressing it (friction will slow the car). Finally, shutting the car off can be used to decrease its speed. Any change in speed or direction means that no object is accelerating.

Describe all the forces acting on a child riding a horse on a merry-go-round. Which of these forces provides the centripetal acceleration of the child?

There are at least three distinct major forces on the child. The force of gravity is acting downward on the child. There is a normal force from the seat of the horse acting upward on the child. There must be friction between the seat of the horse and the child as well, or the child could not be accelerated by the horse. It is that friction that provides the centripetal acceleration. There may be smaller forces as well, such as a reaction force on the child's hands if the child is holding on to part of the horse. Any force that has a radially inward component will contribute to the centripetal acceleration.

A child on a sled comes flying over the crest of a small hill, as shown in Fig. 5-31. His sled does not leave the ground, but he feels the normal force between his chest and the sled decrease as he goes over the hill. Explain this decrease using Newton's second law.

When the child is on a level surface, the normal force between his chest and the sled is equal to the child's weight, and thus he has no vertical acceleration. When he goes over the hill the normal force on him will be reduced. Since the child is moving on a curved path there must be a net centripetal force towards the center of the path, and so the normal force does not completely support the weight. Write newtons second law for the radial direction with inward as positive. We see that the normal force is reduced from mg by the centripetal force.