The universal law of gravitation states that every object in the universe attracts every other object with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The force acts along the line joining the centers of the two objects.
Mathematically: F = G × (m₁ × m₂) / d²
Where F is the gravitational force, G is the universal gravitational constant, m₁ and m₂ are the masses of the two objects, and d is the distance between their centers.
The formula to find the magnitude of gravitational force between the earth and an object on its surface is:
F = G × (M × m) / R²
Where:
F = gravitational force (weight of the object)
G = universal gravitational constant (6.67 × 10⁻¹¹ N m²/kg²)
M = mass of the earth (6 × 10²⁴ kg)
m = mass of the object
R = radius of the earth (6.4 × 10⁶ m)
This force is also equal to the weight of the object: F = m × g, where g = G × M / R² ≈ 9.8 m/s²
Free fall is the motion of an object under the influence of gravitational force only. Whenever objects fall towards the earth under gravitational force alone, without any other force acting on them (like air resistance), they are said to be in free fall.
Acceleration due to gravity is the acceleration experienced by an object when it falls freely under the influence of the earth's gravitational force. It is denoted by 'g' and has a value of approximately 9.8 m/s² on the surface of the earth. This acceleration is always directed towards the center of the earth.
The main differences between mass and weight are:
The weight of an object on the moon is 1/6th of its weight on the earth because:
It is difficult to hold a school bag with thin straps because pressure = force/area. When the strap is thin, the area of contact with the shoulder is small. For the same weight (force), a smaller area results in greater pressure on the shoulder, making it uncomfortable and difficult to carry.
Buoyancy is the upward force exerted by a fluid (liquid or gas) on an object immersed in it. This upward force is called buoyant force or upthrust. It occurs because the pressure at the bottom of the immersed object is greater than the pressure at the top, resulting in a net upward force.
An object floats or sinks in water based on the relative densities of the object and water:
The weighing machine actually measures weight, not mass. It converts weight to mass using the standard value of g (9.8 m/s²). If the actual value of g at your location is different from 9.8 m/s², then your actual mass would be different from 42 kg. However, for most practical purposes, we can consider the measured value as your mass.
The iron bar is actually heavier than the bag of cotton. This is because:
According to the universal law of gravitation, F ∝ 1/d². If the distance is reduced to half (d' = d/2), then:
F' ∝ 1/(d/2)² = 1/(d²/4) = 4/d²
Therefore, F' = 4F
The gravitational force becomes 4 times the original force when the distance is reduced to half.
Although gravitational force is proportional to mass (F ∝ m), acceleration is given by a = F/m. So:
a = F/m = (G M m / d²) / m = G M / d²
This shows that acceleration due to gravity is independent of the mass of the falling object. Therefore, all objects, regardless of their mass, fall with the same acceleration in the absence of air resistance.
Using the formula F = G M m / R²:
F = (6.7 × 10⁻¹¹ N m² kg⁻² × 6 × 10²⁴ kg × 1 kg) / (6.4 × 10⁶ m)²
F = (4.02 × 10¹⁴) / (4.096 × 10¹³) = 9.8 N
The gravitational force between earth and 1 kg object on its surface is 9.8 N.
The earth attracts the moon with the same force as the moon attracts the earth. According to Newton's third law of motion, forces always occur in equal and opposite pairs. The gravitational force between two bodies is equal in magnitude but opposite in direction.
Although the moon attracts the earth with the same force as the earth attracts the moon, the earth does not move significantly toward the moon because:
(i) If mass of one object is doubled: F ∝ m, so force becomes 2F (doubled)
(ii) If distance is doubled: F ∝ 1/d², so force becomes F/4 (one-fourth)
If distance is tripled: F ∝ 1/d², so force becomes F/9 (one-ninth)
(iii) If masses of both objects are doubled: F ∝ m₁m₂, so force becomes 4F (four times)
The universal law of gravitation is important because:
The acceleration of free fall is the acceleration experienced by an object when it falls freely under the influence of gravitational force alone. On earth, this acceleration is approximately 9.8 m/s² and is denoted by 'g'.
The gravitational force between the earth and an object is called the weight of the object.
No, the friend will not agree with the weight of gold bought. This is because:
A sheet of paper falls slower than a crumpled paper ball because:
Mass of object, m = 10 kg
On earth: Weight = m × g = 10 kg × 9.8 m/s² = 98 N
On moon: Weight = (1/6) × weight on earth = (1/6) × 98 N = 16.33 N
Initial velocity, u = 49 m/s
Final velocity at maximum height, v = 0 m/s
Acceleration due to gravity, g = -9.8 m/s² (negative as it opposes motion)
(i) Using v² = u² + 2as
0 = (49)² + 2 × (-9.8) × h
0 = 2401 - 19.6h
19.6h = 2401
h = 2401 / 19.6 = 122.5 m
(ii) Using v = u + at for upward motion
0 = 49 - 9.8t
9.8t = 49
t = 49 / 9.8 = 5 s (time to reach maximum height)
Total time = 2 × 5 = 10 s (time up + time down)
Height, h = 19.6 m
Initial velocity, u = 0 m/s
Acceleration due to gravity, g = 9.8 m/s²
Using v² = u² + 2gh
v² = 0 + 2 × 9.8 × 19.6
v² = 19.6 × 19.6
v = √384.16 = 19.6 m/s
The final velocity just before touching the ground is 19.6 m/s.
Initial velocity, u = 40 m/s
Final velocity at maximum height, v = 0 m/s
Acceleration due to gravity, g = -10 m/s²
Using v² = u² + 2as
0 = (40)² + 2 × (-10) × h
0 = 1600 - 20h
20h = 1600
h = 1600 / 20 = 80 m
Maximum height reached = 80 m
Net displacement = 0 (returns to starting point)
Total distance covered = 80 m (up) + 80 m (down) = 160 m
Using F = G M₁ M₂ / d²
F = (6.7 × 10⁻¹¹ × 6 × 10²⁴ × 2 × 10³⁰) / (1.5 × 10¹¹)²
F = (8.04 × 10⁴⁴) / (2.25 × 10²²)
F = 3.57 × 10²² N
The gravitational force between earth and Sun is 3.57 × 10²² N.
Let the stones meet after time t at height h from ground.
For falling stone:
Distance travelled = 100 - h
Using s = ut + ½gt²
100 - h = 0 × t + ½ × 9.8 × t²
100 - h = 4.9t² ...(1)
For upward thrown stone:
Distance travelled = h
Using s = ut + ½gt² (g = -9.8 m/s²)
h = 25t - ½ × 9.8 × t²
h = 25t - 4.9t² ...(2)
Adding equations (1) and (2):
100 - h + h = 4.9t² + 25t - 4.9t²
100 = 25t
t = 4 s
From equation (2):
h = 25 × 4 - 4.9 × (4)²
h = 100 - 4.9 × 16
h = 100 - 78.4 = 21.6 m
The stones meet after 4 seconds at a height of 21.6 m from the ground.
Total time of flight = 6 s
Time to reach maximum height = 6/2 = 3 s
(a) Using v = u + at at maximum height
0 = u - 9.8 × 3
u = 29.4 m/s
(b) Using v² = u² + 2as
0 = (29.4)² + 2 × (-9.8) × h
19.6h = 864.36
h = 864.36 / 19.6 = 44.1 m
(c) After 4 s, the ball is on its way down
Time from top = 4 - 3 = 1 s
Distance from top = ½ × 9.8 × (1)² = 4.9 m
Position from ground = 44.1 - 4.9 = 39.2 m
The buoyant force on an object immersed in a liquid acts vertically upward, opposite to the direction of gravity.
A block of plastic released under water comes up to the surface because:
Density of substance = mass/volume = 50 g / 20 cm³ = 2.5 g/cm³
Density of water = 1 g/cm³
Since density of substance (2.5 g/cm³) > density of water (1 g/cm³), the substance will sink in water.
Density of packet = mass/volume = 500 g / 350 cm³ = 1.43 g/cm³
Density of water = 1 g/cm³
Since density of packet (1.43 g/cm³) > density of water (1 g/cm³), the packet will sink in water.
When the packet is fully immersed, it will displace water equal to its own volume (350 cm³).
Mass of water displaced = volume × density = 350 cm³ × 1 g/cm³ = 350 g