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A solid sphere of mass M and radius R rolls without slipping down an inclined pl

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Question: 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)

Options:

  1. √(2gh)
  2. √(3gh/2)
  3. √(gh)
  4. √(4gh/3)

Correct Answer: √(2gh)

Exam Year: 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).

A solid sphere of mass M and radius R rolls without slipping down an inclined pl

Practice Questions

Q1
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)
  1. √(2gh)
  2. √(3gh/2)
  3. √(gh)
  4. √(4gh/3)

Questions & Step-by-Step Solutions

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)
  • Step 1: Identify the initial energy of the sphere at the top of the incline. This is the potential energy, which is given by the formula PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.
  • Step 2: Identify the final energy of the sphere at the bottom of the incline. This energy is kinetic energy, which has two parts: translational kinetic energy (due to the motion of the center of mass) and rotational kinetic energy (due to the spinning of the sphere).
  • Step 3: Write the formula for translational kinetic energy, which is KE_trans = (1/2)mv^2, where v is the speed of the center of mass.
  • Step 4: Write the formula for rotational kinetic energy for a solid sphere, which is KE_rot = (1/5)mv^2.
  • Step 5: Combine the two kinetic energy formulas to get the total kinetic energy at the bottom: KE_total = KE_trans + KE_rot = (1/2)mv^2 + (1/5)mv^2.
  • Step 6: Set the initial potential energy equal to the total kinetic energy: mgh = (1/2)mv^2 + (1/5)mv^2.
  • Step 7: Factor out the common term 'm' from both sides of the equation, which simplifies to gh = (1/2)v^2 + (1/5)v^2.
  • Step 8: Combine the fractions on the right side: (1/2 + 1/5) = (5/10 + 2/10) = (7/10), so gh = (7/10)v^2.
  • Step 9: Solve for v^2 by multiplying both sides by (10/7): v^2 = (10/7)gh.
  • Step 10: Take the square root of both sides to find v: v = √(10gh/7).
  • Conservation of Energy – The principle that the total energy in a closed system remains constant, allowing the conversion of potential energy to kinetic energy.
  • Kinetic Energy Components – Understanding that the total kinetic energy includes both translational and rotational components for rolling objects.
  • Rolling Without Slipping – The condition where the sphere rolls down the incline without sliding, affecting the relationship between translational and rotational motion.
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