Chapter 4 question 5

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 M_e and radius R_e). 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.	1
	The explorer is actually weightless.
	There is no any force acting on the explorer.	1

e.	Derive an expression for the speed v_r 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 v_r.	4 marks

	Suppose a satellite of mass m is revolving round the Earth near the surface. The weight of the satellite is mg_o, where g_o is the gravitational field strength on the Earth's surface.
	The centripetal force is fully provided by the weight:
	where R is the Earth's radius.
	The four possible paths are
	*parabola* for v < v_r. *circle* for v = v_r *ellipse* for v_r < v < v_e, where v_e is the escape speed. *hyperbola* for v > v_e, where v_e is the escape speed.

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