Q1. A rotating body has an angular momentum L. If its moment of inertia is doubled and angular velocity is halved, what will be the new angular momentum? (2021)
Solution:
New angular momentum L' = I'ω' = (2I)(ω/2) = L.
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Q2. 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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Q3. A torque of 15 N·m is applied to a wheel with a moment of inertia of 3 kg·m². What is the angular acceleration? (2023)
Solution:
Using τ = Iα, we have α = τ/I = 15/3 = 5 rad/s².
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Q4. A flywheel is rotating with an angular speed of 10 rad/s. If it is brought to rest in 5 seconds, what is the angular deceleration? (2020)
Solution:
Using the formula α = (ω - ω₀)/t, we have α = (0 - 10)/5 = -2 rad/s², so the deceleration is 2 rad/s².
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Q5. A solid cylinder and a hollow cylinder of the same mass and radius are released from rest at the same height. Which one reaches the ground first? (2022)
Solution:
The solid cylinder has a lower moment of inertia, thus it accelerates faster and reaches the ground first.
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Q6. 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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Q7. 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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Q8. A wheel is rotating with an angular velocity of 10 rad/s. If it accelerates at 2 rad/s², what will be its angular velocity after 5 seconds? (2023)
Solution:
Using the formula ω = ω₀ + αt, we have ω = 10 rad/s + (2 rad/s²)(5 s) = 20 rad/s.
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Q9. A uniform rod of length L and mass M is pivoted at one end and released from rest. What is the angular speed of the rod just before it hits the ground? (2019)
Solution:
Using conservation of energy, potential energy at the top converts to rotational kinetic energy at the bottom. The angular speed ω = √(3g/L).
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Q10. 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.
