| a. |
Explain
what is meant by the gravitational field due to a planet. Discuss how the
motion of a satellite round the planet depends on the strength of the gravitational
field. |
4
marks |
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A gravitational field is a region in
which any body that has mass will experience a force of attraction. |
1 |
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The gravitational field strength is the force
per unit mass on a test mass. Suppose a test mass mo
experiences gravitational force F, the field strength at that point
is
 |
1 |
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A satellite orbiting the planet is performing
circular motion, in which the centripetal force is provided by the gravitational
force: |
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where r is the radius of the orbit. |
1 |
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For a particular orbit, the speed of the satellite
is |
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1 |
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| b. |
Write
a statement to define gravitational potential energy. Hence, derive an
expression for the gravitational potential energy of an object on the Earth’s
surface. |
5
marks |
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The gravitational potential energy of an object is defined
as the negative of the work done by the gravitational force as the object
moves from infinity to that point. |
1 |
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Consider a mass m attracted to move from infinity
to r by the Earth. |
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1 |
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The gravitational force acting on m at r
is |
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1 |
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The work done by FG is |
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1 |
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(Since the direction of FG and the increasing
of x are opposite, a negative sign is added:
 |
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Since the gravitational p.e. is negative of W, we
have |
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1 |
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| c. |
What
do you understand by the term “escape speed”? Derive an expression for
the escape speed of a planet in terms of the gravitational field strength
g on the planet’s surface and the radius of the planet, R. |
4
marks |
| |
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|
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The escape speed of a planet is the
minimum projection speed required for any object to escape from the surface
of a planet without return. |
1 |
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For a body to escape from a planet, its initial
kinetic energy must be so large that, at infinity, it is still in motion.
i.e. the total energy at infinity is non-negative. By conservation of energy,
the initial total energy is |
1 |
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1 |
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The escape speed of the planet is |
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1 |
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| d. |
A
spacecraft is raised to the level of an orbit near the Earth’s surface.
Discuss the possible paths taken by a spacecraft for various launching
speeds. |
3
marks |
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The four possible paths are |
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-
parabola when projection speed is very low
-
circle when the projection speed is large enough
for the centripetal force equal to the weight
-
ellipse when the projection speed
is so large that the weight cannot provide all the required centripetal
force
-
hyperbola when the projection speed reaches
the escape speed so that the spacecraft simply does not return
|
2 |
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1 |