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

• The gravitational force increases by a factor of four. According to the law of gravitation, the force is inversely proportional to the square of the distance between two objects. If the distance is halved, the force becomes four times greater.

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

• Both heavy and light objects fall at the same rate because the acceleration due to gravity is constant for all objects, regardless of their masses. The heavier object experiences a greater gravitational force, but it also has more inertia, which balances out the acceleration, resulting in the same falling speed.

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 × 10⁶ m.)

• The magnitude of the gravitational force can be calculated using Newton's law of gravitation:

  $F = \frac{G \times M \times m}{r^2}$

  where:

• $G = 6.67 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2$

• $M = 6 \times 10^{24} \, \text{kg}$

• $m = 1 \, \text{kg}$

• $r = 6.4 \times 10^6 \, \text{m}$

• Substituting the values gives $F \approx 9.8 \, \text{N}$.

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?

• The earth attracts the moon with the same force with which the moon attracts the earth. According to Newton’s third law of motion, every action has an equal and opposite reaction. Therefore, the gravitational forces between the earth and the moon are equal in magnitude and opposite in direction.

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

• The earth does move towards the moon, but due to its much larger mass, the movement is extremely small compared to the movement of the moon towards the earth.

What happens to the force between two objects if:

• (i) The mass of one object is doubled?
  • The gravitational force between the two objects doubles.
• (ii) The distance between the objects is doubled and tripled?
  • If the distance is doubled, the force reduces to one-fourth; if tripled, the force reduces to one-ninth.
• (iii) The masses of both objects are doubled?
  • The gravitational force increases by four times.

What is the importance of the universal law of gravitation?

• The universal law of gravitation explains the force of attraction between all objects with mass, governing the motion of celestial bodies, the structure of galaxies, and various phenomena on Earth, such as the tides.

What is the acceleration of free fall?

• The acceleration of free fall, denoted by $g$, is approximately $9.8 \, \text{m/s}^2$ on the surface of the earth.

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

• The gravitational force between the earth and an object is called weight.

Amit buys a 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.)

• The friend might not agree with the weight of the gold because the value of $g$ (acceleration due to gravity) is greater at the poles than at the equator. This means that the weight of the gold would be slightly less at the equator.

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

• A sheet of paper falls slower than a crumpled ball due to air resistance. The flat sheet has a larger surface area, which increases air resistance, causing it to fall more slowly.

Gravitational force on the surface of the moon is only 1/6 as strong as the gravitational force on the earth. What is the weight in newtons of a 10 kg object on the moon and on the earth?

• On Earth: $F = 10 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 98 \, \text{N}$
• On Moon: $F\_{\text{moon}} = \frac{98 \, \text{N}}{6} \approx 16.3 \, \text{N}$

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

• (i) The maximum height to which it rises:
  • $h = \frac{v^2}{2g} = \frac{49^2}{2 \times 9.8} = 122.5 \, \text{m}$
• (ii) The total time it takes to return to the surface of the earth:
  • Total time = $2 \times \frac{v}{g} = 2 \times \frac{49}{9.8} = 10 \, \text{seconds}$

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

• Final velocity $v = \sqrt{2gh} = \sqrt{2 \times 9.8 \times 19.6} = 19.6 \, \text{m/s}$

A stone is thrown vertically upward with an initial velocity of 40 m/s. Taking $g = 10 \, \text{m/s}^2$, find the maximum height reached by the stone. What is the net displacement and the total distance covered by the stone?

• Maximum height $h = \frac{v^2}{2g} = \frac{40^2}{2 \times 10} = 80 \, \text{m}$
• Net displacement = 0 (since it returns to the original point)
• Total distance = $80 \, \text{m} + 80 \, \text{m} = 160 \, \text{m}$

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

• Using Newton's law of gravitation: $F = \frac{G \times M*{\text{earth}} \times M*{\text{sun}}}{r^2}$
• Substituting the values: $F \approx 3.57 \times 10^{22} \, \text{N}$

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

• Let the stones meet after $t$ seconds at a height $h$ from the ground.
• For the falling stone: $h = 100 - \frac{1}{2} g t^2$
• For the upward stone: $h = v t - \frac{1}{2} g t^2$
• Solving these equations, $t \approx 4 \, \text{seconds}$, and $h \approx 20 \, \text{m}$ from the ground.

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

• (a) The velocity with which it was thrown up:
  • $v = g \times t/2 = 10 \times 3 = 30 \, \text{m/s}$
• (b) The maximum height it reaches:
  • $h = \frac{v^2}{2g} = \frac{30^2}{2 \times 10} = 45 \, \text{m}$
• (c) Its position after 4 s:
  • After 4 seconds, it is on its way down, and has fallen for 1 second. Distance fallen = $\frac{1}{2} \times g \times t^2 = 5 \, \text{m}$ below the max height.

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

• The buoyant force acts in the upward direction, opposite to the force of gravity.

Why does a block of plastic released underwater come up to the surface of the water?

• The block of plastic is less dense than water, so the buoyant force acting on it is greater than the gravitational force, causing it to rise to the surface.

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

• Density of the substance = $\frac{50 \, \text{g}}{20 \, \text{cm}^3} = 2.5 \, \text{g/cm}^3$
• The substance will sink because its density is greater than that of water.

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

• Density of the packet = $\frac{500 \, \text{g}}{350 \, \text{cm}^3} \approx 1.43 \, \text{g/cm}^3$
• The packet will sink because its density is greater than that of water.
• Mass of water displaced = Volume of packet $\times$ Density of water = $350 \, \text{cm}^3 \times 1 \, \text{g/cm}^3 = 350 \, \text{g}$