Gravitation: Potential energy and closest approach of star.
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3.) consider an object of mass m, moving in a circular orbit, subject to a central attractive for e whose magnitude is given by F(r) = h/r^3. a. what are the dimensions of h? b. show that the angular momentum for the motion is uniquely determined by h and m. c. what is the resulting relation b/t period and radius analogous to Kepler's third law for this force?
39.) an object of mass 3x10^15 kg approaches the solar system. When it is very far away, where the gravitational potential energy can be neglected in comparison with its kinetic energy, the object moves w/ a velocity of 12 km/s in a straight line. By straight line extrapolation, the closest this line would come to the sun is 3x10^8 km. the point of the object's nearest approach to the sun is characterized by the fact that the radius vector from the object to the sun is perpendicular to the tangent to the path at that point.
a. sketch the orbit of the object.
b. use conservation of energy and angular momentum to calculate the velocity of the object at the point of nearest approach. c. calculate the distance of nearest approach.
48.) a deep hole in earth reaches a depth of 1/2 of the earth's Radius. How much work is
done when a 1 kg mass is slowly lifted from the bottom of the hole to the earth's surface?
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Solution Summary
The problems solved are about central force, gravitational potential energy, and nearest approach of a star from sun.
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3.) consider an object of mass m, moving in a circular orbit, subject to a central attractive
for e whose magnitude is given by F(r) = h/r^3. a. what are the dimensions of h? b. show
that the angular momentum for the motion is uniquely determined by h and m. c. what is
the resulting relation b/t period and radius analyogous to Kepler's third law for this force?
Answer:
a.
The force F(r) is given by
F = h/r3
Or h = F*r3
Hence the product of dimensions of F and r3 gives the dimensions of h
Or
b.
Let the mass of the object be m and speed of the object is v in the orbit of radius r. As we know that the centripetal force required moving the object in a circular path is given by mv2/r and the central force provides this, we get
F(r) =
This gives
gives ---------------------------- (1)
Now as we know that the angular momentum of an object is the moment of momentum and can be given by m*v*r the angular momentum of the object using equation (1) is given by
Hence the angular momentum of the object is in terms of mass m and the constant h only.
c.
The time period of the object is given by
T = ...
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