Mastering Numerical Problems on Gravitation for Class 9: A Comprehensive Guide

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Summary

Gravitation is a fundamental concept in physics, and understanding the numerical problems associated with it is crucial for Class 9 students. This comprehensive guide delves into the various formulas, principles, and practical applications of gravitation, equipping students with the necessary tools to tackle complex numerical problems with confidence.

Universal Law of Gravitation

numerical problems on gravitation class 9

The Universal Law of Gravitation, formulated by Sir Isaac Newton, states that every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The mathematical expression of this law is:

F = G × (m1 × m2) / r^2

Where:
– F is the gravitational force between the two objects
– G is the gravitational constant, with a value of 6.674 × 10^-11 N⋅m^2/kg^2
– m1 and m2 are the masses of the two objects
– r is the distance between the centers of the two objects

Example Problem:
Two objects with masses of 10 kg and 5 kg are placed 2 meters apart. Calculate the gravitational force between them.

Given:
– m1 = 10 kg
– m2 = 5 kg
– r = 2 m

Substituting the values in the formula:
F = G × (m1 × m2) / r^2
F = (6.674 × 10^-11 N⋅m^2/kg^2) × (10 kg × 5 kg) / (2 m)^2
F = 3.337 × 10^-9 N

Acceleration due to Gravity

The acceleration due to gravity, denoted as ‘g’, is the rate of change of velocity of an object due to the Earth’s gravitational pull. The standard value of acceleration due to gravity on the Earth’s surface is approximately 9.8 m/s^2.

The weight of an object on Earth can be calculated using the formula:

W = m × g

Where:
– W is the weight of the object
– m is the mass of the object
– g is the acceleration due to gravity

Example Problem:
Calculate the weight of an object with a mass of 50 kg on the Earth’s surface.

Given:
– m = 50 kg
– g = 9.8 m/s^2

Substituting the values in the formula:
W = m × g
W = 50 kg × 9.8 m/s^2
W = 490 N

Gravitational Potential Energy

The gravitational potential energy of an object is the energy it possesses due to its position in the gravitational field. The formula to calculate the gravitational potential energy of an object is:

U = m × g × h

Where:
– U is the gravitational potential energy of the object
– m is the mass of the object
– g is the acceleration due to gravity
– h is the height of the object above the reference point (usually the Earth’s surface)

Example Problem:
Calculate the gravitational potential energy of a 20 kg object placed at a height of 5 meters above the ground.

Given:
– m = 20 kg
– g = 9.8 m/s^2
– h = 5 m

Substituting the values in the formula:
U = m × g × h
U = 20 kg × 9.8 m/s^2 × 5 m
U = 980 J

Escape Speed

The escape speed, also known as the escape velocity, is the minimum speed an object must have to break free from the gravitational pull of a planet or other celestial body. The formula to calculate the escape speed is:

v_e = √(2 × g × R)

Where:
– v_e is the escape speed
– g is the acceleration due to gravity
– R is the radius of the planet or celestial body

Example Problem:
Calculate the escape speed from the surface of the Earth, given that the radius of the Earth is approximately 6,371 km and the acceleration due to gravity is 9.8 m/s^2.

Given:
– g = 9.8 m/s^2
– R = 6,371 km = 6,371,000 m

Substituting the values in the formula:
v_e = √(2 × g × R)
v_e = √(2 × 9.8 m/s^2 × 6,371,000 m)
v_e = 11,186 m/s

Orbital Speed

The orbital speed of a satellite or a planet around a celestial body is the speed required to maintain a stable orbit. The formula to calculate the orbital speed is:

v = √(G × M / r)

Where:
– v is the orbital speed
– G is the gravitational constant (6.674 × 10^-11 N⋅m^2/kg^2)
– M is the mass of the central body (e.g., the Earth)
– r is the distance of the satellite or planet from the center of the central body

Example Problem:
Calculate the orbital speed of a satellite orbiting the Earth at a distance of 400 km from the Earth’s surface.

Given:
– G = 6.674 × 10^-11 N⋅m^2/kg^2
– M = 5.972 × 10^24 kg (mass of the Earth)
– r = 6,771,000 m (distance from the Earth’s center)

Substituting the values in the formula:
v = √(G × M / r)
v = √((6.674 × 10^-11 N⋅m^2/kg^2) × (5.972 × 10^24 kg) / (6,771,000 m))
v = 7,611 m/s

Centripetal Force

Centripetal force is the force that keeps an object moving in a circular path. The formula to calculate the centripetal force is:

F_c = m × v^2 / r

Where:
– F_c is the centripetal force
– m is the mass of the object
– v is the velocity of the object
– r is the radius of the circular path

Example Problem:
A 2 kg object is moving in a circular path with a radius of 5 meters and a velocity of 10 m/s. Calculate the centripetal force acting on the object.

Given:
– m = 2 kg
– v = 10 m/s
– r = 5 m

Substituting the values in the formula:
F_c = m × v^2 / r
F_c = 2 kg × (10 m/s)^2 / 5 m
F_c = 40 N

Buoyant Force

The buoyant force is the upward force exerted by a fluid on an object immersed in it. The formula to calculate the buoyant force is:

F_b = ρ × V × g

Where:
– F_b is the buoyant force
– ρ is the density of the fluid
– V is the volume of the object
– g is the acceleration due to gravity

Example Problem:
A 5 kg object is submerged in water (density of water is 1,000 kg/m^3). Calculate the buoyant force acting on the object.

Given:
– ρ = 1,000 kg/m^3 (density of water)
– V = 0.005 m^3 (volume of the object)
– g = 9.8 m/s^2

Substituting the values in the formula:
F_b = ρ × V × g
F_b = 1,000 kg/m^3 × 0.005 m^3 × 9.8 m/s^2
F_b = 49 N

Density

Density is a measure of the mass per unit volume of a substance. The formula to calculate the density of an object is:

ρ = m / V

Where:
– ρ is the density of the object
– m is the mass of the object
– V is the volume of the object

Example Problem:
A 10 kg object has a volume of 2 m^3. Calculate the density of the object.

Given:
– m = 10 kg
– V = 2 m^3

Substituting the values in the formula:
ρ = m / V
ρ = 10 kg / 2 m^3
ρ = 5 kg/m^3

By understanding these formulas and principles, students can effectively tackle a wide range of numerical problems related to gravitation in Class 9. Remember to practice regularly and apply the concepts to real-world scenarios for a deeper understanding of the subject.

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