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Chapter 4 Question 5
a. The gravitational potential energy of a mass m above the Earth’s surface
and at a distance r from the Earth’s center is given by
State the significance of the negative sign in the above equation by
referring to the potential energy when the value of r is very large.
1 mark
The gravitational potential energy for a system of two masses is defined
as zero at infinity, where the force between them is zero.
Since the gravitational force between masses is attractive, to separate
the masses to infinity requires work done on the system. Thus, for any
finite separation, the stored energy is less than that at infinity, i.e. the
gravitational potential energy is negative.
In simple terms, the negative sign in the given equation indicates that
the force between the Earth and any mass is attractive.
1
b. Derive an expression for the total energy of a satellite revolving round the
Earth in a circular orbit of radius r. Using your result, describe how the
motion of the satellite varies if there is net energy loss.
5 marks
Suppose a satellite is orbiting in a circle with radius r at a speed v
around the Earth (mass Me and radius Re). The gravitational force
accounts fully for the centripetal force:
1
The kinetic energy of the satellite is
1
The gravitational p.e. of the satellite is
1
The total energy of the satellite includes both its k.e. and p.e.
1
When there is net energy loss, U decreases. This requires r to decrease.
i.e. the satellite falls towards the Earth.
As r decreases, from equation (1), v increases.
i.e. the satellite moves faster as it falls towards the Earth.
1
c. Explain why an astronaut inside a space shuttle coasting in a circular
orbit round the Earth has a feeling of weightlessness. Compare this to
i)
a passenger inside an aeroplane which is moving parallel to the
Earth’s surface.
6 marks
The astronaut is performing circular motion with the shuttle. His weight
accounts fully for the centripetal force:
1
Since circular motion is an unbalanced motion, the weight needs not be
balanced. Besides his weight, there is no other forces acting on the
astronaut. i.e. there is no force between the astronaut and his seat,
sometimes even no contact.
In fact, we feel our weight because there is a reaction force (normal
contact force) between our body and the ground we stand or the chair
we sit. If such a force vanishes, we would feel weightless.
1
i) Although an aeroplane also moves round the Earth, its motion is
different from that of a space shuttle. The speed of aeroplane is
relatively very slow. The centripetal force is almost zero. Therefore, the
vertical forces are balanced. In fact, the weight of an aeroplane flying
horizontally is balanced by the lift of the air, which can be explain by
using the Bernoulli's principle.
1
The weight of a passenger inside the aeroplane is balanced by the
normal contact force with his chair.
1
ii)
an explorer inside a spaceship coasting in a straight line on
leaving the Solar system, many decades from now.
A spaceship moving in the outer space is free from any gravitational
force.
The explorer is actually weightless.
1
There is no any force acting on the explorer.
1
e. Derive an expression for the speed vr of a satellite revolving round the
Earth in a circular orbit near the Earth’s surface, in terms of the Earth’s
gravitational field strength and its radius. Discuss how the path of the
orbit varies when the launching speed is different from vr.
Suppose a satellite of mass m is revolving round the Earth near the
surface. The weight of the satellite is mgo, where go is the gravitational
field strength on the Earth's surface.
The centripetal force is fully provided by the weight:
4 marks
where R is the Earth's radius.
The four possible paths are
 parabola for v < vr.
 circle
for v = vr
 ellipse
for vr < v < ve, where ve is the escape speed.
 hyperbola for v > ve, where ve is the escape speed.