Physics 1A Week 6 Questions Fall 1998
1. (a) Many people believe that orbiting astronauts feel weightless because they are "beyond the pull of earth's gravity". If a spacecraft were really unaffected by the earth's gravity, would it remain in orbit? Explain. What is the real reason that astronauts in orbit feel "weightless"?
(b) As part of training before going into orbit, astronauts ride in an airliner that is flown along the same parabolic trajectory as a freely falling projectile. Explain why this gives the same feeling of "weightlessness" as being in orbit.
(c) Think about what happens to sky divers when they first jump out of the plane, and when they are falling at terminal speed. When are they "weightless"?
2. Which is correct?
(a) The moon goes around the center of the earth.
(b) The earth goes around the center of the moon.
(c) They both go around some point between their centers.
3. Suppose you can stand on the surface of a star. If the entire star shrinks to 1/3 of its former radius, what will happen to your weight? Suppose that instead of the star shrinking, you tunnel into the star until you are at a distance from the center equal to 1/3 the star's radius. How will your weight change?
4. Why doesn't the gravitational force that acts on a satellite in a circular orbit change its speed? Why does the gravitational force on a satellite in an elliptical orbit change its speed?
5. We know that the relation between orbital speed and radius is given by:
This formula was derived using the inverse-square law. Suppose that gravity instead obeyed an inverse-cube law, i.e. :
In that case, what would the relation between v0 and r be?
Physics 1A Week 6 Solutions Fall 1998
1. (a) You intuitively feel that your weight is the amount with which the ground or chair (or whatever) pushes up on you. If you stand on bathroom scales in an elevator accelerating upwards, you say you feel extra heavy, and the scales read a larger value than usual. If you were in orbit, the scales would read zero. Both you and the scales would be in free fall as you move in a circular path around the center of the earth. Your acceleration towards the center of the circle (v2/r) is the acceleration due to gravity, which is almost as large as it is here on earth. You would be unaware of this acceleration since you and the whole spacecraft have the same acceleration.
(b) The plane is called the "Vomit Comet". All the "weightless" scenes in the movie Apollo 13 movie were filmed in this. As in orbit, the plane experiences an acceleration due to gravity of g, and you and the plane are in free fall together.
(c) When they first jump, they are in free fall, and weightless. When they are at terminal velocity they have no acceleration, and are in equilibrium. The upwards force from air resistance balances their weight. There is an amusement park ride that simulates falling at terminal speed. A large fast fan generates a big stream of upwards flowing air which you can "float" on. The relative velocity between you and the air is the same, but now you are still and the air is moving.
2. (c) They both go around some point between their centers. This point is their center of mass and the position is calculated with a formula similar to the calculation for center of gravity.
For the earth-moon system, this point is inside the earth, at a distance of 4700 km from the earth's center. Note the earth and sun also orbit a common center of mass, which is close to the center of the sun.
3. If the star shrinks its radius decreases, but its mass stays the same. (This can happen! Most stars are made of gas which can be compressed. If the energy source of the sun were turned off, gravity would make the sun shrink.) As the gravitational force for given masses depends on 1/r2, if r decreases to 1/3 its former value, gravitational force increases to nine times its former value.
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Physics 1A Week 6 Solutions Fall 1998
If you were to tunnel into a star (star stays the same size), the gravitational force you experience depends on the amount of mass in a sphere of radius equal to your distance from the star's center, and on your distance from the center. Now the amount of mass in that sphere is proportional to volume, and volume in turn is proportional to r3. So if you are at 1/3 the radius, the amount of volume beneath you is 1/27 as great, and the amount of mass is therefore 1/27 as great (assuming constant density). But you're also 1/3 as far from the center, so that the r2 term in the denominator of the gravitational force law is 1/9 as great. Thus, the net effect of this is that your weight would decrease to 1/3 its former value. What would your weight be if you were right at the center of a star?
4. For motion in a circle at constant speed, the force is always perpendicular to the velocity and towards the center of the circle. That is exactly what happens for a satellite in circular orbit under the influence of gravity. Since there is no component of the gravitational force parallel to the motion, and no other forces are acting, the speed is constant. For an elliptical orbit, the radius changes, thus there is a component of the velocity towards or away from the parent object, in the direction of the gravitational force. Hence in an elliptical orbit, a satellite will go faster as it gets closer to its parent body, and slower as it gets more distant. Comets show this effect dramatically as their orbits are usually very elliptical. Comet Hale-Bopp, for example, has a very elliptical orbit and moves much faster when it is close to the sun. This effect was first quantified by Kepler who deduced the orbit of the planet Mars is an ellipse. It is known as Kepler's Second Law, called the "Rule of Equal Areas".
5. The centripetal force acting on an object in orbit is simply the gravitational force. Therefore,
Solving for v, we get
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