Summary
Numerical problems on gravitation in class 11 involve the application of Newton’s Law of Universal Gravitation to calculate the force of gravity between objects, the gravitational field strength at a given height, and other related quantities. This comprehensive guide provides detailed explanations, formulas, examples, and practice problems to help students master the concepts of gravitation and excel in their class 11 physics examinations.
Understanding Newton’s Law of Universal Gravitation
The foundation of numerical problems on gravitation in class 11 is the understanding of Newton’s Law of Universal Gravitation, which 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 force of gravity between the two objects
– G is the gravitational constant (6.67 × 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 1: Calculating the Force of Gravity
Consider two spheres, each with a mass of 10 kg, separated by a distance of 1 meter. Calculate the force of gravity between the two spheres.
Given:
– m1 = 10 kg
– m2 = 10 kg
– r = 1 m
Substituting the values in the formula:
F = (6.67 × 10^-11 N⋅m^2/kg^2) * (10 kg * 10 kg) / (1 m)^2
F = 6.67 × 10^-9 N
Therefore, the force of gravity between the two spheres is 6.67 × 10^-9 N.
Calculating Gravitational Field Strength
The gravitational field strength, denoted by the symbol ‘g’, is the force per unit mass exerted by the gravitational field on an object. The formula for calculating the gravitational field strength is:
g = G * (m_E) / (r_E + h)^2
Where:
– g is the gravitational field strength
– G is the gravitational constant (6.67 × 10^-11 N⋅m^2/kg^2)
– m_E is the mass of the Earth (5.98 × 10^24 kg)
– r_E is the radius of the Earth (6.371 × 10^6 m)
– h is the height above the surface of the Earth
Example Problem 2: Calculating Gravitational Field Strength at a Given Height
Calculate the gravitational field strength at a height of 4000 m above the surface of the Earth.
Given:
– m_E = 5.98 × 10^24 kg
– r_E = 6.371 × 10^6 m
– h = 4000 m
Substituting the values in the formula:
d = r_E + h = 6.371 × 10^6 m + 4000 m = 6.375 × 10^6 m
g = (6.67 × 10^-11 N⋅m^2/kg^2) * (5.972 × 10^24 kg) / (6.375 × 10^6 m)^2
g = 9.801 m/s^2
Therefore, the gravitational field strength at a height of 4000 m above the surface of the Earth is 9.801 m/s^2.
Numerical Problems on Gravitational Potential Energy
Gravitational potential energy is the potential energy possessed by an object due to its position in a gravitational field. The formula for calculating the gravitational potential energy of an object is:
U = -G * (m1 * m2) / r
Where:
– U is the gravitational potential energy
– G is the gravitational constant (6.67 × 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 3: Calculating Gravitational Potential Energy
Consider two spheres, each with a mass of 5 kg, separated by a distance of 2 meters. Calculate the gravitational potential energy between the two spheres.
Given:
– m1 = 5 kg
– m2 = 5 kg
– r = 2 m
Substituting the values in the formula:
U = -(6.67 × 10^-11 N⋅m^2/kg^2) * (5 kg * 5 kg) / (2 m)
U = -8.34 × 10^-10 J
Therefore, the gravitational potential energy between the two spheres is -8.34 × 10^-10 J.
Numerical Problems on Escape Velocity
The escape velocity is the minimum velocity required for an object to break free from the gravitational pull of a planet or other celestial body. The formula for calculating the escape velocity is:
v_e = √(2G * m / r)
Where:
– v_e is the escape velocity
– G is the gravitational constant (6.67 × 10^-11 N⋅m^2/kg^2)
– m is the mass of the planet or celestial body
– r is the radius of the planet or celestial body
Example Problem 4: Calculating Escape Velocity
Calculate the escape velocity from the surface of the Earth, given that the mass of the Earth is 5.98 × 10^24 kg and the radius of the Earth is 6.371 × 10^6 m.
Given:
– m = 5.98 × 10^24 kg
– r = 6.371 × 10^6 m
Substituting the values in the formula:
v_e = √(2 * 6.67 × 10^-11 N⋅m^2/kg^2 * 5.98 × 10^24 kg / 6.371 × 10^6 m)
v_e = 11.2 km/s
Therefore, the escape velocity from the surface of the Earth is 11.2 km/s.
Numerical Problems on Gravitational Potential
Gravitational potential is the potential energy per unit mass of an object in a gravitational field. The formula for calculating the gravitational potential is:
V = -G * m_E / r
Where:
– V is the gravitational potential
– G is the gravitational constant (6.67 × 10^-11 N⋅m^2/kg^2)
– m_E is the mass of the Earth (5.98 × 10^24 kg)
– r is the distance from the center of the Earth
Example Problem 5: Calculating Gravitational Potential at a Given Height
Calculate the gravitational potential at a height of 4000 m above the surface of the Earth.
Given:
– G = 6.67 × 10^-11 N⋅m^2/kg^2
– m_E = 5.98 × 10^24 kg
– r = 6.371 × 10^6 m + 4000 m = 6.375 × 10^6 m
Substituting the values in the formula:
V = -(6.67 × 10^-11 N⋅m^2/kg^2) * (5.98 × 10^24 kg) / (6.375 × 10^6 m)
V = -6.23 × 10^6 J/kg
Therefore, the gravitational potential at a height of 4000 m above the surface of the Earth is -6.23 × 10^6 J/kg.
Numerical Problems on Gravitational Potential Energy of a System
The gravitational potential energy of a system is the sum of the gravitational potential energies of all the objects in the system. The formula for calculating the gravitational potential energy of a system is:
U_system = -G * Σ(m_i * m_j) / r_ij
Where:
– U_system is the gravitational potential energy of the system
– G is the gravitational constant (6.67 × 10^-11 N⋅m^2/kg^2)
– m_i and m_j are the masses of the objects in the system
– r_ij is the distance between the centers of the objects
Example Problem 6: Calculating Gravitational Potential Energy of a Three-Object System
Consider a system of three objects with masses 5 kg, 10 kg, and 15 kg, separated by distances of 2 m, 3 m, and 4 m, respectively. Calculate the gravitational potential energy of the system.
Given:
– m1 = 5 kg, m2 = 10 kg, m3 = 15 kg
– r12 = 2 m, r13 = 3 m, r23 = 4 m
Substituting the values in the formula:
U_system = -(6.67 × 10^-11 N⋅m^2/kg^2) * [(5 kg * 10 kg) / 2 m + (5 kg * 15 kg) / 3 m + (10 kg * 15 kg) / 4 m]
U_system = -1.67 × 10^-9 J
Therefore, the gravitational potential energy of the three-object system is -1.67 × 10^-9 J.
Numerical Problems on Gravitational Potential Energy of a Satellite
The gravitational potential energy of a satellite in orbit around a planet or celestial body is given by the formula:
U = -G * (m_s * m_p) / r
Where:
– U is the gravitational potential energy of the satellite
– G is the gravitational constant (6.67 × 10^-11 N⋅m^2/kg^2)
– m_s is the mass of the satellite
– m_p is the mass of the planet or celestial body
– r is the distance between the centers of the satellite and the planet/celestial body
Example Problem 7: Calculating Gravitational Potential Energy of a Satellite
A satellite with a mass of 500 kg is orbiting the Earth at a distance of 7000 km from the center of the Earth. Calculate the gravitational potential energy of the satellite.
Given:
– m_s = 500 kg
– m_p = 5.98 × 10^24 kg (mass of the Earth)
– r = 7000 km + 6371 km (radius of the Earth) = 13371 km = 1.3371 × 10^7 m
Substituting the values in the formula:
U = -(6.67 × 10^-11 N⋅m^2/kg^2) * (500 kg * 5.98 × 10^24 kg) / (1.3371 × 10^7 m)
U = -3.33 × 10^9 J
Therefore, the gravitational potential energy of the satellite is -3.33 × 10^9 J.
Numerical Problems on Gravitational Potential Energy of a Falling Object
The gravitational potential energy of an object falling in a gravitational field is given by the formula:
U = m * g * h
Where:
– U is the gravitational potential energy of the object
– m is the mass of the object
– g is the gravitational field strength
– h is the height from which the object is falling
Example Problem 8: Calculating Gravitational Potential Energy of a Falling Object
An object with a mass of 10 kg is dropped from a height of 50 m above the surface of the Earth. Calculate the gravitational potential energy of the object at the moment of release.
Given:
– m = 10 kg
– g = 9.81 m/s^2 (gravitational field strength at the surface of the Earth)
– h = 50 m
Substituting the values in the formula:
U = 10 kg * 9.81 m/s^2 * 50 m
U = 4905 J
Therefore, the gravitational potential energy of the object at the moment of release is 4905 J.
Numerical Problems on Gravitational Potential Energy of a System of Falling Objects
The gravitational potential energy of a system of falling objects is the sum of the gravitational potential energies of all the objects in the system. The formula for calculating the gravitational potential energy of a system of falling objects is:
U_system = Σ(m_i * g * h_i)
Where:
– U_system is the gravitational potential energy of the system
– m_i is the mass of the i-th object
– g is the gravitational field strength
– h_i is the height from which the i-th object is falling
Example Problem 9: Calculating Gravitational Potential Energy of a System of Falling Objects
Consider a system of three objects with masses 5 kg, 10 kg, and 15 kg, dropped from heights of 20 m, 30 m, and 40 m, respectively. Calculate the gravitational potential energy of the system.
Given:
– m1 = 5 kg, m2 = 10 kg, m3 = 15 kg
– h1 = 20 m, h2 = 30 m, h3 = 40 m
– g = 9.81 m/s^2
Substituting the values in the formula:
U_system = (5 kg * 9.81 m/s^2 * 20 m) + (10 kg * 9.81 m/s^2 * 30 m) + (15 kg * 9.81 m/s^2 * 40 m)
U_system = 981 J + 2943 J + 5886 J
U_system = 9810 J
Therefore, the gravitational potential energy of the system of three falling objects is 9810 J.
Reference:
- Gravitation and Satellite Physics by Richard P. Feynman
- Gravitation Numericals by Resonance Eduventures
- Universal Law of Gravitation from Physics LibreTexts
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