F.6 Gravitation

Eternal Snow

2010-11-12 7:20 pm

1.) The mass ratio of the two planets X and Y is k:1 and their separation is d. THe position with the gravitational potential at a maximum us at a distance r from X, find an expression for r

2.)The velocity of escape from the earth is V. For a planet with radius four times that of the earth and with density three times that of the earth, the velocity of escape from the planet would be,

3.) A student suggests that gravity can be thought of in the following way:
'Gravity behaves like an invisible, elastic 'sponge' surrounding the earth. When a ball is thrown up in the air it fallls back to the earth because the 'sponge' is compressed and forces the ball back downwards. Things like rockets can escape from the earth because they have enough energy to break right through the 'sponge'
A ball is thrown vertically into the air. Sketch the variation of the velocity of the ball with time during its upward and downward motion according to (a) student's sponge theory (b) Newton's theory

Please give me detail explanation. Thanks!

回答 (1)

天同

2010-11-14 7:00 pm

✔ 最佳答案

1. Since gravitational force F = -dV/dr, where dV/dr is the potential energy gradient

At the distance r from the planet X, the potential is maximum, this indicates that the force at r is zero.

Hence, G(km)m'/r^2 = Gmm'/(d-r)^2where G is the Universal Gravitational Constant and m' is the mass of a small object at distance r.

k/r^2 = 1/(d-r)^2
solve for r gives r ={ -2kd + 2d.square-root[k(2k-1)] }/2(k-1)

2. Escape velocity V = square-root[2GM/R]
where G is the Universal Gravitational Constant, M is the mass of the earth and R is its radius.

For the planet, we have its mass M' = (4/3).pi(4R)^3.(3d)
where d is the density of earth and pi = 3.14159.....
hence, M' = [(4/3).pi.R^3.d] x (192) = 192M

Thus, the escape velocity of the planet
= square-root[2G x 192M/4R] = square-root[2GM/R x 48]
= 6.93 x square-root[2GM/R] = 6.93V

3. If gravity behaves like an elastic sponge, then gravity follows Hooke's Law, i.e. F = kh, where h is the height from earth surface.
In that case, the variation of velocity of the stone with time follows a cosine curve (from 0 to 180 degrees)

If using Newton's Law, assuming the height reached by the stone is small such that the variation of the acceleration due to gravity g is neligible, the velocity of the stone then varies linearly with time.
The curve is a straght line with -ve slope and a +ve y-intercept.

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