Q2. A satellite is in a circular orbit around the Earth. What is the relationship between the gravitational force and the centripetal force acting on the satellite? (2022)
Solution:
For a satellite in a stable orbit, the gravitational force provides the necessary centripetal force, hence they are equal.
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Q3. A wheel of radius R and mass M is rolling without slipping on a horizontal surface. If it has a linear speed v, what is its total kinetic energy? (2022)
Solution:
The total kinetic energy is the sum of translational and rotational kinetic energy. K.E. = (1/2)Mv² + (1/2)(Iω²) where I = (1/2)MR² for a solid cylinder.
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Q4. A uniform rod of length L and mass M is pivoted at one end and released from rest. What is the angular velocity of the rod just before it hits the ground? (2019)
Solution:
Using conservation of energy, potential energy at the top is converted to rotational kinetic energy at the bottom. The angular velocity ω can be found using the relation ω = √(3g/L).
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Q5. A solid sphere of mass M and radius R rolls without slipping down an inclined plane of height h. What is the speed of the center of mass of the sphere at the bottom of the incline? (2021)
Solution:
Using conservation of energy, potential energy at the top = kinetic energy at the bottom. The total kinetic energy is the sum of translational and rotational kinetic energy. Thus, mgh = (1/2)mv^2 + (1/5)mv^2, leading to v = √(10gh/7).
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Q6. A satellite is in a circular orbit around the Earth. What is the expression for the centripetal force acting on the satellite? (2022)
Solution:
All the given expressions represent the centripetal force acting on the satellite in different forms.
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Q7. What is the angular momentum of a particle of mass 2 kg moving in a circle of radius 3 m with a speed of 4 m/s? (2021)
Solution:
Angular momentum L = mvr, where v is the tangential speed. L = 2 kg * 4 m/s * 3 m = 24 kg·m²/s.
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Q8. A solid cylinder and a hollow cylinder of the same mass and radius are rolling down an incline. Which one reaches the bottom first? (2023)
Solution:
The solid cylinder has a smaller moment of inertia compared to the hollow cylinder, thus it accelerates faster and reaches the bottom first.
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Q9. If the angular momentum of a system is conserved, what can be said about the net external torque acting on the system? (2023)
Solution:
If angular momentum is conserved, it implies that the net external torque acting on the system is zero.
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Q10. A rotating body has an angular momentum L. If the moment of inertia of the body is I, what is the angular velocity ω of the body? (2021)
Solution:
Angular momentum L is given by L = Iω, thus ω = L/I.
