Chapter 09 Gravitation Exercise

Exercises

1. How does the force of gravitation between two objects change when the distance between them is reduced to half?

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Answer

Consider the Universal law of gravitation,

According to that law, the force of attraction between two bodies is

F=(Gm1m2)r2

Where,

m1 and m2 are the masses of the two bodies.

G is the gravitational constant.

r is the distance between the two bodies.

Given that the distance is reduced to half then,

r=1/2 r

F=(Gm1m2)r2

F=(Gm1m2)(r2)2

F=4(Gm1m2)r2

F=4F

Therefore once the space between the objects is reduced to half, then the force of gravitation will increase by fourfold the first force.

2. Gravitational force acts on all objects in proportion to their masses. Why then, a heavy object does not fall faster than a light object?

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Answer

All objects fall from the top with a constant acceleration called acceleration due to gravity (g). This is constant on earth and therefore the value of ‘g’ doesn’t depend on the mass of an object. Hence, heavier objects don’t fall quicker than light-weight objects provided there’s no air resistance.

3. What is the magnitude of the gravitational force between the earth and a 1kg object on its surface? (Mass of the earth is 6×1024kg and radius of the earth is 6.4×106m.)

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Answer

From Newton’s law of gravitation, we know that the force of attraction between the bodies is given by

F=(Gm1m2)r2

Here

m1= mass of Earth =6.0×1024 kgm2= mass of the body =1 kg

r= distance between the two bodies

Radius of Earth =6.4×106 m

G= Universal gravitational constant =6.67×1011Nm2 kg2

By substituting all the values in the equation

F=(Gm1m2)r2F=6.67×1011(6.0×1024×1)(6.4×106)2 F=9.8 N

This shows that Earth exerts a force of 9.8 N on a body of mass 1 kg. The body will exert an equal force of attraction of 9.8 N on the Earth.

4. The earth and the moon are attracted to each other by gravitational force. Does the earth attract the moon with a force that is greater or smaller or the same as the force with which the moon attracts the earth? Why?

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Answer

The earth attracts the moon with a force same as the force with which the moon attracts the earth. However, these forces are in opposite directions. By universal law of gravitation, the force between moon and also the sun can be

F=(Gm1m2)d2

Where,

d= distance between the earth and moon.

m1 and m2= masses of earth and moon respectively.

5. If the moon attracts the earth, why does the earth not move towards the moon?

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Answer

According to the universal law of gravitation and Newton’s third law, we all know that the force of attraction between two objects is the same, however in the opposite directions. So the earth attracts the moon with a force same as the moon attracts the earth but in opposite directions. Since earth is larger in mass compared to that of the moon, it accelerates at a rate lesser than the acceleration rate of the moon towards the Earth. Therefore, for this reason the earth does not move towards the moon.

6. What happens to the force between two objects, if

(i) the mass of one object is doubled?

(ii) the distance between the objects is doubled and tripled?

(iii) the masses of both objects are doubled?

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Answer

(i)

According to universal law of gravitation, the force between 2 objects (m1 and m2) is proportional to their plenty and reciprocally proportional to the sq. of the distance(R) between them.

F=(G2m1m2)R2

If the mass is doubled for one object.

F=2 F, so the force is also doubled.

(ii)

If the distance between the objects is doubled and tripled

If it’s doubled

Hence,

F=(Gm1 m2)/(2R)2 F=1/4(Gm1 m2)/R2 F=F/4

Force thus becomes one-fourth of its initial force.

Now, if it’s tripled

Hence,

F=(Gm1 m2)/(3R)2 F=1/9(Gm1 m2)/R2 F=F/9

Force thus becomes one-ninth of its initial force.

(iii)

If masses of both the objects are doubled, then

F=(G2m12m2)R2

F=4F, Force will therefore be four times greater than its actual value.

7. What is the importance of universal law of gravitation?

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Answer

The universal law of gravitation explains many phenomena that were believed to be unconnected:

(i) The motion of the moon round the earth

(ii) The responsibility of gravity on the weight of the body which keeps us on the ground

(iii) The tides because of the moon and therefore the Sun

(iv) The motion of planets round the Sun

8. What is the acceleration of free fall?

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Answer

Acceleration due to gravity is the acceleration gained by an object due to gravitational force. On Earth, all bodies experience a downward force of gravity which Earth’s mass exerts on them. The Earth’s gravity is measured by the acceleration of the freely falling objects. At Earth’s surface, the acceleration of gravity is 9.8 ms2 and it is denoted by ’ g ‘. Thus, for every second an object is in free fall, its speed increases by about 9.8 metres per second.

9. What do we call the gravitational force between the earth and an object?

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Answer

The gravitation force between the earth and an object is called weight. Weight is equal to the product of acceleration due to the gravity and mass of the object.

10. Amit buys few grams of gold at the poles as per the instruction of one of his friends. He hands over the same when he meets him at the equator. Will the friend agree with the weight of gold bought? If not, why? [Hint: The value of g is greater at the poles than at the equator.]

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Answer

The weight of a body on the earth’s surface;

W = mg (where m = mass of the body and g = acceleration due to gravity)

The value of g is larger at poles when compared to the equator. So gold can weigh less at the equator as compared to the poles.

Therefore, Amit’s friend won’t believe the load of the gold bought.

11. Why will a sheet of paper fall slower than one that is crumpled into a ball?

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Answer

A sheet of paper has a larger surface area when compared to a crumpled paper ball. A sheet of paper will face a lot of air resistance. Thus, a sheet of paper falls slower than the crumpled ball.

12. Gravitational force on the surface of the moon is only 16 as strong as gravitational force on the earth. What is the weight in newtons of a 10kg object on the moon and on the earth?

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Answer

Given data:

Acceleration due to earth’s gravity =gc or g=9.8 m/s2

Object’s mass, m=10 kg

Acceleration due to moon gravity =gm

Weight on the earth =Wc

Weight on the moon =Wm

Weight = mass × gravity

gm=(1/6)gc (given)

So Wm=mgm=mx(1/6)gc

Wm=10×(1/6)×9.8=16.34 N

Wc=mxgc=10×9.8

We=98 N

13. A ball is thrown vertically upwards with a velocity of 49m/s. Calculate

(i) the maximum height to which it rises,

(ii) the total time it takes to return to the surface of the earth.

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Answer

Given data:

Initial velocity u=49 m/s

Final speed v at maximum height =0

Acceleration due to earth gravity g=9.8 m/s2 (thus negative as ball is thrown up).

By third equation of motion,

2gH=v2u22×(9.8)×H=0(49)219.6H=2401H=122.5 m

Total time T= Time to ascend (Ta)+ Time to descend (Td)

v=u+gt0=49+(9.8)×TaTa=(49/9.8)=5 s

Also, Td=5 s

Therefore T=Ta+Td

T=5+5 T=10 s

14. A stone is released from the top of a tower of height 19.6m. Calculate its final velocity just before touching the ground.

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Answer

Given data:

Initial velocity

u=0

Tower height = total distance =19.6 m

g=9.8 m/s2

Consider third equation of motion

v2=u2+2gsv2=0+2×9.8×19.6v2=384.16v=(384.16)v=19.6 m/s

15. A stone is thrown vertically upward with an initial velocity of 40m/s. Taking g=10m/s2, find the maximum height reached by the stone. What is the net displacement and the total distance covered by the stone?

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Answer

Given data:

Initial velocity u=40 m/s

g=10 m/s2

Max height final velocity =0

Consider third equation of motion

v2=u22gs [negative as the object goes up]

0=(40)22×10×s s=(40×40)/20

Maximum height s=80 m

Total Distance =s+s=80+80

Total Distance =160 m

Total displacement =0 (The first point is the same as the last point)

16. Calculate the force of gravitation between the earth and the Sun, given that the mass of the earth =6×1024kg and of the Sun =2×1030kg. The average distance between the two is 1.5×1011m.

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Answer

Given data:

Mass of the sun ms=2×1030 kg

Mass of the earth mc=6×1024 kg

Gravitation constant G=6.67×1011 N m2/kg2

Average distance r=1.5×1011 m

Consider Universal law of Gravitation

F=(Gm1m2)d2F=(6.67×1011×6×1024×2×1030)(1.5×1011)2 F=3.56×1022 N

17. A stone is allowed to fall from the top of a tower 100m high and at the same time another stone is projected vertically upwards from the ground with a velocity of 25m/s. Calculate when and where the two stones will meet.

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Answer

Given data:

(i) When the stone from the top of the tower is thrown, Initial velocity u=0

Distance travelled =x

Time taken =t

Therefore,

s=ut+12gt2x=0+(1/2)g2x=5t2(a)

(ii) When the stone is thrown upwards,

Initial velocity u=25 m/s

Distance travelled =(100x)

Time taken =t

s=ut12gt2(100x)=25t(1/2)×10xt2x=10025t+5t2 (b) 

From equations (a) and (b)

5t2=10025t+5t2t=(100/25)=4sec.

After 4sec, two stones will meet

From (a)

x=5t2=5x4×4=80 m.

Putting the value of x in (100-x)

=(10080)=20 m

This means that after 4sec,2 stones meet a distance of 20 m from the ground.

18. A ball thrown up vertically returns to the thrower after 6s. Find

(a) the velocity with which it was thrown up,

(b) the maximum height it reaches, and

(c) its position after 4s.

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Answer

Given data:

g=10 m/s2

Total time T=6sec

Ta=Td=3sec (a) Final velocity at maximum height v=0

From first equation of motion:-

v=ugtau=v+gta=0+10×3=30 m/s

The velocity with which stone was thrown up is 30 m/s.

(b) From second equation of motion

s=uta12g(ta)2=30×3(1/2)×10×(3)2=9045=45 m

The maximum height stone reaches is 45 m.

(c) In 3sec, it reaches the maximum height.

Distance travelled in another 1sec=s,

s=uta12g(ta)2 s=0+10×1×1 s=5 m.

The distance travelled in another 1sec=5 m.

Therefore in 4sec, the position of point p(455) =40 m from the ground.

19. In what direction does the buoyant force on an object immersed in a liquid act?

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Answer

The buoyant force on an object that is immersed in a liquid will be in a vertically upward direction.

20. Why does a block of plastic released under water come up to the surface of water?

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Answer

The density of plastic is lesser than that of water. Therefore, the force of buoyancy on plastic block will be greater than the weight of plastic block. Hence, the acceleration of plastic block is going to be in the upward direction. So, the plastic block comes up to the surface of water.

21. The volume of 50g of a substance is 20cm3. If the density of water is 1gcm3, will the substance float or sink?

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Answer

To find the Density of the substance the formula is

Density =( Mass/Volume )

Density =(50/20)=2.5 g/cm3

Density of water =1 g/cm3

Density of the substance is greater than density of water. So the substance will sink.

22. The volume of a 500g sealed packet is 350cm3. Will the packet float or sink in water if the density of water is 1g cm3 ? What will be the mass of the water displaced by this packet?

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Answer

Density of sealed packet =500/350=1.42 g/cm3

Density of sealed packet is greater than density of water

Therefore the packet will sink.

Considering Archimedes Principle,

Displaced water volume = Force exerted on the sealed packet.

Volume of water displaced =350 cm3

Therefore displaced water mass =ρ×V

=1×350

Mass of displaced water =350 g.



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