14. What is the magnitude of the gravitational force between the earth and a 1 kg object on its surface?
(Mass of the earth is 6 × 1024 kg and radius of the earth is 6.4 × 106m.)
Solution:
From Newton’s law of gravitation, we know that the force of attraction between the bodies is given by
\(F = \frac{(Gm_1m_2)}{r^2}\)
m1 = mass of Earth = 6.0 x 1024 kg
m2 = mass of the body = 1kg
r = distance between the two bodies
Radius of Earth = 6.4 x 106 m
G = Universal gravitational constant = 6.67 x 10-11 Nm2 kg2
By substituting all the values in the equation
\(F = \frac{(Gm_1m_2)}{r^2}\)
\(F = \frac{6.67 \times 10^{-11}(6.0 \times 10^{24} \times 1)}{(6.4 \times 10^6 )^2}\)
\(F = 9.8 N\)
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.
15. 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?
Solution:
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
\(F = \frac{(Gm_1m_2)}{d^2}\)
Where,
d = distance between the earth and moon.
m1 and m2 = masses of earth and moon respectively.
16. If the moon attracts the earth, why does the earth not move towards the moon?
Solution:
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.
17. 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?
Solution:
(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.
\(F = \frac{(G\,2m_1m_2)}{R^2}\)
If the mass is doubled for one object.
F = 2F, so the force is also doubled.
(ii) If the distance between the objects is doubled and tripled
If it’s doubled
Hence,
F = (Gm1m2)/(2R)2
F = 1/4 (Gm1m2)/R2
F = F/4
Force thus becomes one-fourth of its initial force.
Now, if it’s tripled
Hence,
F = (Gm1m2)/(3R)2
F = 1/9 (Gm1m2)/R2
F = F/9
Force thus becomes one-ninth of its initial force.
(iii) If masses of both the objects are doubled, then
\(F = \frac{(G\,2m_12m_2)}{R^2}\)
F = 4F, Force will therefore be four times greater than its actual value.
18. What is the importance of universal law of gravitation?
Solution:
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
19. What is the acceleration of free fall?
Solution:
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 ms-2 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.
20. What do we call the gravitational force between the earth and an object?
Solution:
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.
21. 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.]
Solution:
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.
22. Why will a sheet of paper fall slower than one that is crumpled into a ball?
Solution:
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.
23. Gravitational force on the surface of the moon is only 1/6 as strong as gravitational force on the earth. What is the weight in newton’s of a 10 kg object on the moon and on the earth?
Solution:
Given data:
Acceleration due to earth’s gravity = ge or g = 9.8 m/s2
Object’s mass, m = 10 kg
Acceleration due to moon gravity = gm
Weight on the earth= We
Weight on the moon = Wm
Weight = mass x gravity
gm = (1/6) ge (given)
So Wm = m gm = m x (1/6) ge
Wm = 10 x (1/6) x 9.8 = 16.34 N
We = m x ge = 10 x 9.8
We = 98N
