Q#1
Can two particles be in equilibrium under the action of their mutual gravitational force? Can three particles be? Can one of the three particles be?Answer:
If stationary, the net force on each of the two particles will not be zero, and they will exert the gravitational force on each other. So, they will move towards each other and not be in equilibrium. If moving around their center of mass, they may maintain the relative distance as the gravitational force on each of the particles is balanced by the centrifugal force and the net force is zero.
Again for the same reason, three particles cannot be in equilibrium if stationary. If moving, they may maintain their relative distances for some configurations, again the gravitational force is balanced by centrifugal force.
One of the three particles can be in equilibrium even without moving. This situation can happen if two of the particles are moving around the center of mass and the third particle is stationary at the center of mass.
Q#2
Is there any meaning of "Weight of the earth"?
Answer:
No. Because the weight is the force by which the earth attracts a body. So the weight of the earth itself has no meaning.
Q#3
If heavier bodies are attracted more strongly by the earth, why don't they fall faster than the lighter bodies?
Answer:
The heavier bodies are attracted more strongly by the earth means the force applied by the earth is more. This force is the weight of the body = mg. Hence the acceleration of the body = force/mass = mg/m = g, which is independent of m. So all bodies fall with the same acceleration g.
Q#4
Can you think of two particles which do not exert a gravitational force on each other?
Answer:
No. Since the particles have mass, they will exert a gravitational force on each other.
Q#5
The earth revolves round the sun because the sun attracts the earth. The sun also attracts the moon and this force is about twice as large as the attraction of the earth on the moon. Why does the moon not revolve round the sun? Or does it?
Answer:
The moon too revolves around the sun along with the earth.
Q#6
At noon, the sun and the earth pull the objects on the earth's surface in opposite directions. At midnight the sun and the earth pull these objects in the same direction. Is the weight of an object, as measured by a spring balance on the earth's surface, more at midnight as compared to its weight at noon?
Answer:
The distance of the sun from the earth is R = 150 x 10⁶ km while the diameter of the earth is 12800 km only. So, the variance of distance R between the noon and the midnight is comparatively very small (about 0.0085%). The force on an object by the sun is inversely proportional to R², so the change in the force between the noon and the midnight is even smaller or say negligible. So, the weight measured by a spring balance at the earth's surface at these two times will not differ as the spring balances are not sensitive enough to detect such negligible changes.
Q#7
An apple falls from a tree. An insect in the apple finds that the earth is falling towards it with an acceleration g. Who exerts the force needed to accelerate the earth with this acceleration g?
Answer:
From the observation of the insect, his frame of reference is noninertial. With the frame moving with an acceleration g, we will have to apply a pseudo force Mg on the earth in the opposite direction of g, if we want to use Newton's laws in this noninertial frame. So from the insect's view, this force is being exerted by the apple on the earth though it is not correct.
Q#8
Suppose the gravitational potential due to a small system is k/r² at a distance r from it. What will be the gravitational field? Can you think of any such system? What happens if there were negative masses?
Answer:
The gravitational potential V = k/r².
Hence gravitational field E = -dV/dr = -(-2k/r³) = 2k/r³
A hypothetical gravitational dipole system can be assumed like this, as in the case of an electric dipole in which the electric field is inversely proportional to the cube of the distance. It is hypothetical because in the case of gravitation negative masses do not occur.
If there were negative masses the direction of the field would be reversed.
Q#9
The gravitational potential energy of a two-particle system is derived in this chapter as U = -Gm₁m₂/r. Does it follow from the equation that the potential energy for r = ∞ must be zero? Can we choose the potential energy for r = ∞ to be 20 J and still use this formula? If no, what formula should be used to calculate the gravitational potential energy at separation r?
Answer:
Yes. Since U is inversely proportional to r, at r = ∞ the gravitational potential energy must be zero.
If we choose the potential energy at r = ∞ to be 20 J, we can not use this formula, because putting r = ∞, will not give the result 20 J.
The required formula will be,
U = (-Gm₁m₂/r) + 20.
Q#10
The weight of an object is more at the poles than at the equator. Is it beneficial to purchase goods at the equator and sell them at the poles? Does it matter whether a spring balance is used or an equal beam balance is used?
Answer:
The weight of an object is about 0.50% (Very roughly) more at poles than at the equator. Though it seems beneficial at a glance to purchase goods at the equator and sell them at the poles, there are many factors which prove just the opposite. First, the input cost will become much higher than 0.50% in transporting them to a distant (9000 + km) and difficult place like poles. Second, there is no population on the poles, hence no purchaser and no market. Again the change in weight is noticeable only in a spring balance. No change in weight will be there if the equal beam balance is used because it compares the weight of an object to a standard weight and the increase in weight at poles will be same for the object and the standard weight.
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