End of chapter exercises
Problem 1:
An object attracts another with a gravitational force F. If the distance between the centres of
the two objects is now decreased to a third (13) of the original distance, the force of
attraction that the one object would exert on the other would become...
1. 19F
2. 13F
3. 3F
4. 9F
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Answer 1:9F
Problem 2:
An object is dropped from a height of 1 km above the Earth. If air resistance is ignored, the
acceleration of the object is dependent on the...
1. mass of the object
2. radius of the earth
3. mass of the earth
4. weight of the object
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Answer 2:
weight of the object
Problem 3:
A man has a mass of 70 kg on Earth. He is walking on a new planet that has a mass four
times that of the Earth and the radius is the same as that of the Earth
( ME=6×1024 kg, rE=6×106 m )
1. Calculate the force between the man and the Earth.
2. What is the man's weight on the new planet?
3. Would his weight be bigger or smaller on the new planet? Explain how you arrived at
your answer.
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Answer 3:
1. FE=Gm1m2d2=(6,67×10−11)(70)(6×1024)(6×106)2=778,2 N
2. The mass of the new planet is four times the mass of the Earth so we get:
Fplanet=Gm14m2d2=4FE=(778,2)(4)=3112,8 N
3. His weight is bigger on the new planet. The new planet has the same radius as the
Earth, but a larger mass. Since the mass of the planet is proportional to the force of
gravity (or the weight) the man's weight must be larger.
Problem 4:
Calculate the distance between two objects, 5000 kg and 6 × 1012 kg respectively, if the
magnitude of the force between them is 3 × 108 N
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Answer 4:
F3×1083×108d2d2=Gm1m2d2=(6,67×10−11)(5000)(6×1012)d2=2,001×106=6,67×10−3=
8,2×10−2 m
Problem 5:
An astronaut in a satellite 1600 km above the Earth experiences a gravitational force of
magnitude 700 N on Earth. The Earth's radius is 6400 km. Calculate:
1. The magnitude of the gravitational force which the astronaut experiences in the
satellite.
2. The magnitude of the gravitational force on an object in the satellite which weighs 300
N on Earth.
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Answer 5:
1. We first work out the mass of the astronaut:
F7009,77mAmA=Gm1m2d2=(6,67×10−11)(6×1024)mA(6400×103)2=700=71,6 kg
Now we can work out the gravitational force in the satellite:
FS=Gm1m2d2=(6,67×10−11)(6×1024)(71,6)(6400×103+1600×103)2=448 N
2. We first work out the mass of the object:
F3009,77mOmO=Gm1m2d2=(6,67×10−11)(6×1024)mO(6400×103)2=300=30,7 kg
Now we can work out the gravitational force in the satellite:
FS=Gm1m2d2=(6,67×10−11)(6×1024)(30,7)(6400×103+1600×103)2=192 N
Problem 6:
An astronaut of mass 70 kg on Earth lands on a planet which has half the Earth's radius
and twice its mass. Calculate the magnitude of the force of gravity which is exerted on him
on the planet.
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Answer 6:
The gravitational force on Earth is:
FE=GmEmAr2E
On the planet we have twice the Earth's mass and half the Earth's radius:
FP=GmPmAr2P=2GmEmAr2E4=8FE=8(70)(9,8)=5488 N
Problem 7:
Calculate the magnitude of the gravitational force of attraction between two spheres of lead
with a mass of 10 kg and 6 kg respectively if they are placed 50 mm apart.
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Answer 7:
F=Gm1m2d2=(6,67×10−11)(10)(6)(50×10−3)2=1,6×106 N
Problem 8:
The gravitational force between two objects is 1200 N. What is the gravitational force
between the objects if the mass of each is doubled and the distance between them halved?
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Answer 8:
The gravitational force is:
F1=Gm1m2d2
If we double each mass and halve the distance between them we now have:
F2=G(2m1)(2m2)(0,5d)2=4Gm1m20,25d2=16F1
So the force will be 16 times as much.
Problem 9:
Calculate the gravitational force between the Sun with a mass of 2 × 1030 kg and the Earth
with a mass of 6 × 1024kg if the distance between them is 1,4 × 108 km.
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Answer 9:
F=Gm1m2d2=(6,67×10−11)(2×1030)(6×1024)(1,4×1011)2=4,1×1022 N
Problem 10:
How does the gravitational force of attraction between two objects change when
1. the mass of each object is doubled.
2. the distance between the centres of the objects is doubled.
3. the mass of one object is halved, and the distance between the centres of the objects
is halved.
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Answer 10:
1. The gravitational force will be four times as much.
2. The gravitational force will be one fourth as much or four times smaller.
3. The gravitational force will be twice as much.
Problem 11:
Read each of the following statements and say whether you agree or not. Give reasons for
your answer and rewrite the statement if necessary:
1. The gravitational acceleration g is constant.
2. The weight of an object is independent of its mass.
3. G is dependent on the mass of the object that is being accelerated.
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Answer 11:
1. Agree
2. Disagree. Weight is related to mass via the gravitational acceleration
3. Disagree. G is a universal constant
Problem 12:
An astronaut weighs 750 N on the surface of the Earth.
1. What will his weight be on the surface of Saturn, which has a mass 100 times greater
than the Earth, and a radius 5 times greater than the Earth?
2. What is his mass on Saturn?
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Answer 12:
1. On Earth we have:
aE=GMER2E
On Saturn we note that MS=100ME and RS=5RE. So the gravitational acceleration on
Saturn is:
aS∴aS=GMSR2S=G100ME25R2E=4GMER2E=4aE
So the weight of the astronaut on Saturn is:
aS=4aE=4(750)=3000 N
2. On Earth his mass is:
aEm=mgE=aEgE=7509,8=76,53 kg
His mass on Saturn is the same as his mass on Earth, it is only his weight that is different.
Problem 13:
Your mass is 60 kg in Paris at ground level. How much less would you weigh after taking a
lift to the top of the Eiffel Tower, which is 405 m high? Assume the Earth's mass is 6,0 ×
1024 kg and the Earth's radius is 6400 km.
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Answer 13:
We start with your weight on the surface of the Earth. The gravitational acceleration at the
surface of the Earth is 9,8 m·s−2 and so your weight is:
Fg=mg=(60)(9,8)=588 N
At the top of the Eiffel Tower the gravitational acceleration is:
ao=GMEarthd2=(6,67×10−11)(6,0×1024)(6400×103+405)2=9,77 m⋅s−2
Your weight is:
Fg=mg=(60)(9,77)=586,2 N
So you would weigh 1,8 N less.
Problem 14:
1. State Newton's law of universal gravitation.
2. Use Newton's law of universal gravitation to determine the magnitude of the
acceleration due to gravity on the Moon.
The mass of the Moon is 7,4 × 1022 kg.
The radius of the Moon is 1,74 × 106 m.
3. Will an astronaut, kitted out in his space suit, jump higher on the Moon or on the Earth?
Give a reason for your answer.
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Answer 14:
1. Every body in the universe exerts a force on every other body. The force is directly
proportional to the product of the masses of the bodies and inversely proportional to
the square of the distance between them.
2. g=Fmo=Gmmoonmor2m2=Gmmoonr2=(6,67×10−11)(7,4×1022)(1,74×106)2=1,63
3. He will be able to jump higher on the moon. The acceleration due to gravity is lower on
the moon than on the Earth and so there is less gravitational force pulling him down on
the moon.