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Chapter 4 Question 2
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
A gravitational field is a region in which any body that has mass will
experience a force of attraction.
1
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
A satellite orbiting the planet is performing circular motion, in which the
centripetal force is provided by the gravitational force:
where r is the radius of the orbit.
1
For a particular orbit, the speed of the satellite is
1
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
5 marks
surface.
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
Consider a mass m attracted to move from infinity to r by the Earth.
1
The gravitational force acting on m at r is
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The work done by FG is
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(Since the direction of FG and the increasing of x are opposite, a
negative sign is added:
Since the gravitational p.e. is negative of W, we have
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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
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
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
1
The escape speed of the planet is
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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
The four possible paths are
 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
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1