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AL Physics/Gravitation/P.1
PLK VICWOOD K.T. CHONG SIXTH FORM COLLEGE
FORM SIX AL PHYSICS EXERCISE
Gravitation
Given : Gravitational Constant G = 6.67 × 10-11 Nm2kg-2
1. A small, uniform sphere of mass 0.100 kg is
located on the line between a 5.00 kg and a
10.0 kg uniform sphere (Figure 1). The small
sphere is 4.00 m to the right of the center of the
5.00 kg sphere and 6.00 m to the left of the
center of the 10.0 kg sphere. What are the
magnitude and direction of the force on the
0.100 kg sphere? [2.3 × 10-13 N, to the left]
2. Two spheres, each of mass 6.40 kg, are fixed
at points A and B (Figure 2). Find the
magnitude and direction of the initial
acceleration of a sphere with mass 0.010 kg if it
is released from rest at point P and acted on
only forces of gravitational attraction of the
spheres at A and B. [5.1 × 10-8 m/s2,
downward along line midway between A and
B]
Figure 1
10.0 kg
5.00 kg
0.100 kg
4.00 m
6.00 m
P
Figure 2
0.010 kg
10. 0 cm
10.0 cm
6.0 cm
A
B
8.0 cm
6.40 kg
8.0 cm
6.40 kg
3. Two point masses are located at (x, y) coordinates as follows: 5.00 kg at (1.00 m, 0) and 3.00 kg
at (0, -0.500 m). What will be the magnitude and direction of the gravitational force on a 0.0100
kg test mass placed at the origin? [8.68 × 10-12 N; 67.4° clockwise from +x-axis]
4. How far from the earth must a body be along a line toward the sun so that the sun's gravitational pull
balances the earth's? The sun is 1.5 × 108 km away and its mass is 3.24 × 105 Me. [2.6 × 105 km]
5. (a) With what horizontal speed must a satellite be projected at 160 km above the surface of the
earth so that it will have a circular orbit about the earth? Take the earth's radius as 6400 km and
mass as 6 × 1024 kg. (b) What will be the period of revolution? [(a) 7810 m/s (b) 1.47 hr]
6. Assume that the moon orbits the earth in a circular orbit. From the observed orbital period of 27.3
days, calculate the distance of the moon from the center of the earth. Assume that the motion of the
moon is determined solely by the gravitational force exerted on it by the earth. Mass of earth = 6 ×
1024 kg. [3.83 × 108 m]
7. An earth satellite moves in a circular orbit with an orbital speed of 6400 m/s. (a) Find the time of
one revolution. (b) Find the radial acceleration of the satellite in its orbit. Mass of earth = 6 × 1024
kg. [(a) 159 min. (b) 4.20 m/s2]
8. Pluto orbits the sun in a nearly circular orbit with radius 5.90 × 1012 m and an orbital period of
247.7 years. Use these data to calculate the mass of the sun. [1.99 × 1030 kg]
9. What is the period of revolution of an artificial satellite of mass m that is orbiting the earth in a
circular path of radius 8380 km? Mass of earth = 6 × 1024 kg. [2.1 hr]
AL Physics/Gravitation/P.2
10. The mean distance of Mars from the sun is 1.52 times that of Earth from the sun. Find the number
of years required for Mars to make one revolution about the sun. [1.87 yr.]
11. If a satellite in an elliptical orbit about the earth has a perigee (closest distance of approach) of 300
km above the surface of the earth, and an apogee (furthest distance of approach) of 2000 km
above the surface of the earth, then what is the ratio of the orbital speed at perigee to that at
apogee? Take the earth's radius as 6400 km. [1.25]
12. Three point masses are fixed at positions shown in Figure
12. (a) calculate the x- and y-components of the
gravitational field at point P, which is at the origin of coordinates, due to these three point masses. (b) What would
be the magnitude and direction of the force on a 0.0200-kg
mass placed at P? [(a) x-component: 4.55 × 10-10 N/kg;
y-component: 4.55 × 10-10 N/kg (b) 1.3 × 10-11 N, along
the diagonal]
1.0 kg
2.0 kg
Figure 12
0.50 m
P
0.50 m
1.0 kg
13. At what distance above the surface of the earth is the magnitude of its gravitational field 4.90 m/s2 if
the gravitational field at the surface has magnitude 9.8 m/s2? Given that the radius of the earth is
6400 km. [2.65 × 106 m]
14. What is the period of a pendulum (period = 2 s on the earth) on the surface of the moon? The
moon’s mass is 7.35 × 1022 kg and its radius is 1,720 km. Given that g on the earth is 9.8 m/s2.
[4.9 s]
15. The acceleration due to gravity at the north pole of Jupiter is approximately 25 m/s2. Jupiter has
mass 1.9 × 1027 kg and radius 7.1 × 107 m and makes one revolution about its axis in about 10 hr.
(a) What is the gravitational force on a 5.0 kg object at the north pole of Jupiter? (b) What is the
apparent weight of this same object at the equator of Jupiter? [(a) 125 N (b) 114.2 N]
16. If a pendulum has a period of exactly one second at the Equator, what would be its period at the
South Pole? Given that the radius of the earth is 6400 km. [0.998 s]
17. An 800-kg mass and a 600-kg mass are separated by 0.25 m. What is the gravitational potential
due to these masses at a point 0.20 m from the 800-kg mass and 0.15 m from the 600-kg mass? [5.34 × 10-7 J/kg]
18. Masses of 200 and 800 g are 12 cm apart. (a) Find the gravitational force on an object of unit
mass situated at a point on the line joining the masses 4.0 cm from the 200-g mass. (b) Find the
gravitational potential energy per unit mass at that point. (c) How much work is needed to move
this object to a point 4.0 cm from the 800-g mass along the line of centres? [(a) Zero. (b) -10-9
J/kg. (c) -5.0 × 10-10 J.]
19. The asteroid Toro, discovered in 1964, has a radius of about 5.0 km and mass of about 2.0 × 1015
kg. Calculate the escape speed for an object at the surface of Toro. [7.3 m/s]
20. Mars has a mean diameter of 6900 km, Earth one of 1.3 × 104 km. The mass of Mars is 0.11 ME.
(a) How does the mean density of Mars compare with that of Earth? (b) What is the value of g on
AL Physics/Gravitation/P.3
Mars? (c) What is the escape velocity on Mars? Given that g on the earth is 9.8 m/s2. [(a) ρM =
0.74 ρE. (b) 3.85 m/s2. (c) 5.2 km/s.]
21. A projectile is fired vertically from the earth’s surface with an initial speed of 10 km/s. Neglecting
atmospheric friction, how far above the surface of the earth would it go? Take the earth’s radius as
6400 km and the gravity g on the surface of the earth as 9.8 m/s2. [2.6 × 104 km]
22. The Earth is elliptical with polar and equatorial radii equal to 6.357 × 106 m and 6.378 × 106 m
respectively. Estimate the difference, ∆g, in the values of the acceleration of free fall at a pole and
at the equator due to this difference in radii. What other factors affect ∆g? Given : Mass of Earth
= 6 × 1024 kg; Gravitational constant, G = 6.67 × 10-11 Nm2kg-2. [0.065 m/s2]
23. A man is able to jump vertically 1.5 m on earth. What height might he be expected to jump on a
planet of which the density is one third that of the earth but of which the radius is one half that of the
earth? [9 m]
24. A planet X of mass m = 7.35 × 1022 kg and radius R1 = 1.74 × 106 m revolves in a circular orbit
about a star Y of mass M = 5.98 × 1024 kg and radius R2 = 6.63 × 106 m. The period of revolution
is 27.3 days.
V
Figure 24b
+
R
Y
0
P
R2
X
R2
r0
R - R1
-
r
R1
Figure 24a
(a) Find the radius of the orbit, R.
(b) P is a point on the line joining the centres of X and Y as shown in Figure 24a. Figure 24b shows
the variation of the gravitational potential V at point P as a function of r for R2 ≤ r ≤ R - R1,
where r is the distance between P and the centre of Y. V is at a maximum where r = r0.
In your answer book, use the information shown in Figure 24b to sketch F, the gravitational
force per unit mass, as a function of r for R2 ≤ r ≤ R - R1, and put down the value of F at r = r0
on your sketch. Hence find r0.
(c) Explain briefly how
(i) the kinetic energy of X, and
(ii) the radius of the orbit,
would be affected if the air resistance force on X caused by the atmosphere of Y were small but
not negligible, and the orbit were nearly circular.
[(a) 3.83 × 108 m (b) 3.45 × 108 m]
r
AL Physics/Gravitation/P.4
25. This question poses a number of short problems about the dynamics of the motion of a spacecraft.
The mass of the spacecraft is 2000 kg. Figure 25 shows its orbit, and is to scale (except for the
diameters of the Earth and Moon which are shown larger than they should be). The arrows on the
orbit show the direction of motion. Distances given are to the centre of the Earth. Where
calculations are needed, explain how you make each calculation. [Given: Mass of Earth = 6 × 1024
kg; Mass of Moon = 7.35 × 1022 kg]
Note: The orbit is shown as it would appear if the Earth were fixed in space, and all calculations
may make the same assumption.
radius atE
20 000 km
Earth
Moon
380 000 km
(Earth to Moon)
A
E
B
200 000 km
20°
100 000 km
D
C
Figure 25
v = 6 000 m/s
(a) At point A the spacecraft has zero acceleration when its thrust motor is not switched on.
Suggest a reason why.
(b) At point B, again without any motor thrust, the speed of the spacecraft is constant, though it
does have an acceleration. (At B the craft is travelling at right angles to the line joining it to the
Moon’s centre.) Explain this observation.
(c) At C, the velocity is 6000 m/s, as shown. If it were required to alter course by an angle of
1/100 radian, without changing the speed, for how long must a motor thrust of 12000 N be
maintained? In what direction must the thrust be?
(d) In what direction is the spacecraft accelerating at D, if the thrust motor is not in use? Calculate
the magnitude of the change of its momentum over a time of 100 seconds. Explain, together
with a labelled sketch, how you would work out the momentum mv 2 of the spacecraft after this
time, if you knew the momentum mv 1 before it. You need not make the calculation. Ignore any
effect of the Moon, in this part and in part (f) onwards.
(e) Make the rough calculation to show that it is far to ignore the effect of the Moon in (d).
(f) The thrust motors are not used between C and E. By how much does the gravitational potential
energy of the spacecraft change between C and E? Explain whether it is an increase or a
decrease, and whether the answer would have been different if the motors had been used.
(g) At C the spacecraft’s velocity is 6000 m/s. Calculate the kinetic energy of the spacecraft at E,
using the answer to (f).
(h) What kinetic energy would the spacecraft have at E if, instead of having come in from Moon
orbit, the spacecraft were in permanent circular orbit around the Earth, through E?
[(c) 10 s, perpendicular to the velocity (d) 8 × 103 kg·ms-1 (f) -3.6 × 1010 J (g) 7.2 × 1010 J
(h) 2 × 1010 J]