GRAVITATION - Q&A

1. State the universal law of gravitation.

Answer: The universal law of gravitation states that every object in the universe attracts every other object with a force which is proportional to the product of their masses and inversely proportional to the square of the distance between them. The force is along the line joining the centres of two objects.

2. Write the formula to find the magnitude of the gravitational force between the earth and an object on the surface of the earth.

Answer: The formula is:
F = G (M × m) / R²
Where:
F = Gravitational force
G = Universal gravitational constant
M = Mass of the earth
m = Mass of the object
R = Radius of the earth (distance between the center of the earth and the object)

Questions on Page 104

1. What do you mean by free fall?

Answer: Whenever objects fall towards the earth under the influence of the earth's gravitational force alone, without any other external forces (like air resistance) acting on them, we say that the objects are in free fall.

2. What do you mean by acceleration due to gravity?

Answer: When an object falls towards the earth, there is a change in its velocity due to the gravitational force of the earth. This change in velocity involves acceleration. This acceleration produced by the gravitational force of the earth is called acceleration due to gravity. It is denoted by 'g'.

Questions on Page 106

1. What are the differences between the mass of an object and its weight?

Answer:
1. Mass: Mass is the measure of inertia of an object. It is the quantity of matter contained in it. Mass is constant and does not change from place to place. Its SI unit is kilogram (kg).
2. Weight: Weight is the force with which the earth attracts an object. It depends on the acceleration due to gravity (W = m × g), so it varies from place to place. Its SI unit is Newton (N).

2. Why is the weight of an object on the moon 1/6th its weight on the earth?

Answer: The weight of an object depends on the gravitational force exerted by the celestial body. The mass of the moon is less than that of the earth, and its radius is also different. The acceleration due to gravity on the moon (g_m) is approximately one-sixth of the acceleration due to gravity on the earth (g_e). Since Weight = mass × g, the weight of an object on the moon becomes 1/6th of its weight on the earth.

Questions on Page 109

1. Why is it difficult to hold a school bag having a strap made of a thin and strong string?

Answer: The force exerted by the bag is its weight. Pressure is defined as Force divided by Area (P = F/A). A thin string has a very small cross-sectional area. Therefore, the weight of the bag acting on this small area exerts a very large pressure on the shoulder. This high pressure causes pain and makes it difficult to hold the bag.

2. What do you mean by buoyancy?

Answer: When an object is immersed in a fluid (liquid or gas), the fluid exerts an upward force on the object. This upward force is known as upthrust or the force of buoyancy.

3. Why does an object float or sink when placed on the surface of water?

Answer: An object floats or sinks based on the difference between its density and the density of water:
- If the density of the object is less than the density of water, the buoyant force (upthrust) exerted by water is greater than the weight of the object, so it floats.
- If the density of the object is greater than the density of water, the buoyant force is less than the weight of the object, so it sinks.

Questions on Page 110

1. You find your mass to be 42 kg on a weighing machine. Is your mass more or less than 42 kg?

Answer: Your mass is slightly more than 42 kg.
Explanation: A weighing machine measures the weight (normal force) which is calibrated to display mass. When you stand on the machine, the air exerts a small upward buoyant force on your body. This reduces the reading slightly. The actual weight equals the reading plus the buoyant force. Therefore, the actual mass is slightly higher than the reading shown.

2. You have a bag of cotton and an iron bar, each indicating a mass of 100 kg when measured on a weighing machine. In reality, one is heavier than other. Can you say which one is heavier and why?

Answer: The bag of cotton is heavier in reality.
Explanation: Both show a reading of 100 kg, meaning the net downward force (True Weight - Buoyant Force) is the same for both. Cotton has a much lower density than iron, so for the same mass, cotton has a much larger volume. A larger volume displaces more air, resulting in a larger upward buoyant force on the cotton.
True Weight = Measured Weight + Buoyant Force.
Since the buoyant force on cotton is greater than on iron, the true weight (and thus true mass) of the cotton is greater.

Exercises

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

Answer: The gravitational force F is inversely proportional to the square of the distance d (F ∝ 1/d²).
If the distance is reduced to half (d' = d/2), the new force F' becomes:
F' ∝ 1/(d/2)²
F' ∝ 1/(d²/4)
F' ∝ 4/d²
F' = 4F
So, the force becomes 4 times the original 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?

Answer: The acceleration due to gravity (g) is given by the formula g = GM/R², where M is the mass of the earth and R is the radius of the earth. This formula does not include 'm' (the mass of the falling object). Therefore, acceleration due to gravity is independent of the mass of the object. Hence, all objects, whether heavy or light, fall with the same acceleration and speed (ignoring air resistance).

3. 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.)

Answer:
Given:
Mass of earth (M) = 6 × 10²⁴ kg
Mass of object (m) = 1 kg
Radius of earth (R) = 6.4 × 10⁶ m
Gravitational constant (G) = 6.67 × 10^-11 N m² kg^-2
Formula: F = G (M × m) / R²
F = (6.67 × 10^-11 × 6 × 10²⁴ × 1) / (6.4 × 10⁶)²
F = (40.02 × 10¹³) / (40.96 × 10¹²)
F = 40.02 × 10 / 40.96
F = 400.2 / 40.96
F ≈ 9.8 N

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?

Answer: The earth attracts the moon with the same force with which the moon attracts the earth.
Reason: According to Newton's Third Law of Motion, for every action, there is an equal and opposite reaction. The gravitational force is a mutual interaction pair; the force exerted by A on B is equal in magnitude and opposite in direction to the force exerted by B on A.

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

Answer: According to Newton's Second Law of Motion, Acceleration = Force / Mass (a = F/m). While the force exerted by the moon on the earth is large, the mass of the earth is extremely huge. Consequently, the acceleration produced in the earth is negligibly small and unnoticeable. Therefore, we do not see the earth moving 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?

Answer: Formula: F ∝ (m₁ × m₂) / d²
(i) If the mass of one object is doubled (say m₁ becomes 2m₁): The force is directly proportional to mass, so the force doubles (2F).
(ii) If distance is doubled (d becomes 2d): F ∝ 1/(2d)² = 1/4d². The force becomes one-fourth (F/4).
If distance is tripled (d becomes 3d): F ∝ 1/(3d)² = 1/9d². The force becomes one-ninth (F/9).
(iii) If masses of both objects are doubled (2m₁ and 2m₂): The product of masses becomes 4 times (2 × 2 = 4). The force becomes 4 times (4F).

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

Answer: The universal law of gravitation explains several phenomena:
1. It explains the force that binds us to the earth.
2. It explains the motion of the moon around the earth.
3. It explains the motion of planets around the Sun.
4. It explains the tides formed by the rising and falling of water level in the ocean due to the moon and the Sun.

8. What is the acceleration of free fall?

Answer: The acceleration of free fall is the acceleration experienced by an object falling freely towards the earth due to gravitational force. Its value near the earth's surface is approximately 9.8 m/s².

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

Answer: The gravitational force between the earth and an object is called the weight of the object (or simply gravity).

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.]

Answer: No, the friend will not agree with the weight.
Reason: Weight is given by W = m × g. The shape of the earth is not a perfect sphere; it is flattened at the poles. Consequently, the value of 'g' is greater at the poles than at the equator. Therefore, the same mass of gold will weigh more at the poles and less at the equator. When the friend weighs it at the equator, the reading will be less than what Amit bought at the poles.

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

Answer: This happens due to air resistance. A sheet of paper has a larger surface area compared to the crumpled ball. As a result, the sheet experiences a larger upward resistance force from the air, which opposes its motion and slows it down. The crumpled ball, having a smaller surface area, experiences less air resistance and falls faster.

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

Answer:
Mass (m) = 10 kg
Acceleration due to gravity on earth (g_e) = 9.8 m/s²
Weight on Earth (W_e):
W_e = m × g_e = 10 × 9.8 = 98 N

Weight on Moon (W_m):
Since gravity on moon is 1/6th of earth:
W_m = 1/6 × W_e
W_m = 98 / 6
W_m ≈ 16.33 N

13. A ball is thrown vertically upwards with a velocity of 49 m/s. Calculate
(i) the maximum height to which it rises,
(ii) the total time it takes to return to the surface of the earth.

Answer:
Given: Initial velocity (u) = 49 m/s, Final velocity at max height (v) = 0, g = 9.8 m/s² (acting downwards, so a = -9.8 m/s²).
(i) Maximum height (s):
Using equation: v² - u² = 2as
0² - (49)² = 2 × (-9.8) × s
-2401 = -19.6 × s
s = 2401 / 19.6
s = 122.5 m

(ii) Total time (t):
Time to reach top (t₁):
v = u + at
0 = 49 + (-9.8) × t₁
9.8 t₁ = 49
t₁ = 49 / 9.8 = 5 s
Since time of ascent = time of descent, Total time = 2 × 5 = 10 s.

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

Answer:
Given: Initial velocity (u) = 0 (released), height (s) = 19.6 m, g = 9.8 m/s².
Using equation: v² - u² = 2as
v² - 0 = 2 × 9.8 × 19.6
v² = 19.6 × 19.6
v = √(19.6 × 19.6)
v = 19.6 m/s

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

Answer:
Given: u = 40 m/s, v = 0 (at top), g = 10 m/s² (upward motion, so a = -10 m/s²).
Maximum Height (h):
v² - u² = 2as
0 - (40)² = 2 × (-10) × h
-1600 = -20 h
h = 1600 / 20 = 80 m

Net Displacement:
Since the stone goes up and falls back to the starting point, the final position coincides with the initial position.
Net Displacement = 0 m

Total Distance:
Total distance = Distance up + Distance down
Total distance = 80 m + 80 m = 160 m

16. 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.

Answer:
Given:
M_earth = 6 × 10²⁴ kg
M_sun = 2 × 10³⁰ kg
Distance (d) = 1.5 × 10¹¹ m
G = 6.67 × 10^-11 N m² kg^-2
Formula: F = G (M_e × M_s) / d²
F = (6.67 × 10^-11 × 6 × 10²⁴ × 2 × 10³⁰) / (1.5 × 10¹¹)²
F = (6.67 × 12 × 10^(-11+24+30)) / (2.25 × 10²²)
F = (80.04 × 10⁴³) / (2.25 × 10²²)
F = (80.04 / 2.25) × 10^(43-22)
F ≈ 35.57 × 10²¹ N
F ≈ 3.56 × 10²² N

17. 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.

Answer:
Let them meet after time 't'.
For the falling stone (Stone A):
u = 0, a = g. Distance covered (d₁) = ut + ½gt² = ½gt²
For the projected stone (Stone B):
u = 25 m/s, a = -g. Distance covered (d₂) = ut + ½at² = 25t - ½gt²
Total distance:
d₁ + d₂ = Height of tower
½gt² + (25t - ½gt²) = 100
25t = 100
t = 4 s (They meet after 4 seconds)

Where they meet:
Height from ground (d₂) = 25(4) - ½(9.8)(4)²
d₂ = 100 - ½(9.8)(16)
d₂ = 100 - 78.4
d₂ = 21.6 m from the ground (or 78.4 m from the top)

18. A ball thrown up vertically returns to the thrower after 6 s. Find
(a) the velocity with which it was thrown up,
(b) the maximum height it reaches, and
(c) its position after 4 s.

Answer:
Total time = 6 s. Since time of ascent = time of descent, Time to reach top (t) = 3 s.
g = 9.8 m/s². At max height, v = 0.
(a) Initial velocity (u):
v = u + at
0 = u + (-9.8) × 3
u = 29.4 m/s

(b) Maximum height (h):
s = ut + ½at² (Using t=3)
h = 29.4(3) + ½(-9.8)(3)²
h = 88.2 - 44.1
h = 44.1 m

(c) Position after 4 s:
t = 4 s
s = ut + ½at²
s = 29.4(4) + ½(-9.8)(4)²
s = 117.6 - 4.9(16)
s = 117.6 - 78.4
s = 39.2 m (from the ground)

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

Answer: The buoyant force on an object immersed in a liquid acts vertically upwards.

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

Answer: The density of plastic is less than the density of water. Therefore, the upward buoyant force exerted by water on the plastic block is greater than the downward weight of the block. This net upward force pushes the block to the surface.

21. The volume of 50 g of a substance is 20 cm³. If the density of water is 1 g cm^-3, will the substance float or sink?

Answer:
Mass of substance = 50 g
Volume of substance = 20 cm³
Density of substance = Mass / Volume = 50 / 20 = 2.5 g cm^-3
Density of water = 1 g cm^-3
Since the density of the substance (2.5 g cm^-3) is greater than the density of water (1 g cm^-3), the substance will sink.

22. 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 cm^-3? What will be the mass of the water displaced by this packet?

Answer:
Mass of packet = 500 g
Volume of packet = 350 cm³
Density of packet = Mass / Volume = 500 / 350 ≈ 1.43 g cm^-3
Density of water = 1 g cm^-3
Since the density of the packet is greater than the density of water, the packet will sink.

Mass of water displaced:
Since the packet sinks, it will displace a volume of water equal to its own volume (350 cm³).
Mass of water displaced = Volume of water displaced × Density of water
Mass of water displaced = 350 cm³ × 1 g/cm³ = 350 g.