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If a solid cylinder rolls without slipping, what is the ratio of its translation

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Question: If a solid cylinder rolls without slipping, what is the ratio of its translational kinetic energy to its rotational kinetic energy?

Options:

  1. 1:1
  2. 2:1
  3. 1:2
  4. 3:1

Correct Answer: 1:2

Solution: 

For a solid cylinder, the translational kinetic energy (KE_trans) is (1/2)mv² and the rotational kinetic energy (KE_rot) is (1/2)(Iω²). The ratio KE_trans:KE_rot is 1:2.

If a solid cylinder rolls without slipping, what is the ratio of its translation

Practice Questions

Q1
If a solid cylinder rolls without slipping, what is the ratio of its translational kinetic energy to its rotational kinetic energy?
  1. 1:1
  2. 2:1
  3. 1:2
  4. 3:1

Questions & Step-by-Step Solutions

If a solid cylinder rolls without slipping, what is the ratio of its translational kinetic energy to its rotational kinetic energy?
  • Step 1: Understand that a solid cylinder rolling without slipping has both translational and rotational motion.
  • Step 2: Identify the formula for translational kinetic energy (KE_trans), which is KE_trans = (1/2)mv², where m is mass and v is velocity.
  • Step 3: Identify the formula for rotational kinetic energy (KE_rot), which is KE_rot = (1/2)(Iω²), where I is the moment of inertia and ω is the angular velocity.
  • Step 4: For a solid cylinder, the moment of inertia I is (1/2)mr², where r is the radius of the cylinder.
  • Step 5: Relate the translational velocity v to the angular velocity ω using the formula v = rω.
  • Step 6: Substitute ω in the rotational kinetic energy formula: KE_rot = (1/2)(I)(v/r)².
  • Step 7: Replace I with (1/2)mr² in the KE_rot formula: KE_rot = (1/2)((1/2)mr²)(v/r)².
  • Step 8: Simplify the expression for KE_rot to find that KE_rot = (1/4)mv².
  • Step 9: Now, find the ratio of KE_trans to KE_rot: KE_trans:KE_rot = (1/2)mv² : (1/4)mv².
  • Step 10: Simplify the ratio: (1/2) : (1/4) = 2 : 1.
  • Step 11: Therefore, the ratio of translational kinetic energy to rotational kinetic energy is 1:2.
  • Translational Kinetic Energy – The energy due to the motion of the center of mass of the cylinder, calculated as (1/2)mv².
  • Rotational Kinetic Energy – The energy due to the rotation of the cylinder about its axis, calculated as (1/2)(Iω²), where I is the moment of inertia and ω is the angular velocity.
  • Rolling Without Slipping – A condition where the point of contact between the rolling object and the surface does not slide, linking translational and rotational motion.
  • Moment of Inertia – A measure of an object's resistance to changes in its rotation, for a solid cylinder, I = (1/2)mr².
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