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A solid cylinder of radius R rolls down a frictionless incline. What is the rati

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Question: A solid cylinder of radius R rolls down a frictionless incline. What is the ratio of its translational kinetic energy to its total kinetic energy at the bottom?

Options:

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

Correct Answer: 2:1

Solution: 

At the bottom, total kinetic energy = translational + rotational. For a solid cylinder, the ratio of translational to total kinetic energy is 2:1.

A solid cylinder of radius R rolls down a frictionless incline. What is the rati

Practice Questions

Q1
A solid cylinder of radius R rolls down a frictionless incline. What is the ratio of its translational kinetic energy to its total kinetic energy at the bottom?
  1. 1:1
  2. 2:1
  3. 1:2
  4. 3:1

Questions & Step-by-Step Solutions

A solid cylinder of radius R rolls down a frictionless incline. What is the ratio of its translational kinetic energy to its total kinetic energy at the bottom?
  • Step 1: Understand that a solid cylinder is rolling down an incline without friction.
  • Step 2: Recognize that when the cylinder reaches the bottom, it has two types of kinetic energy: translational (movement of the center of mass) and rotational (spinning around its axis).
  • Step 3: Know that the total kinetic energy (KE_total) at the bottom is the sum of translational kinetic energy (KE_trans) and rotational kinetic energy (KE_rot).
  • Step 4: For a solid cylinder, the formula for translational kinetic energy is KE_trans = (1/2)mv^2, where m is mass and v is velocity.
  • Step 5: The formula for rotational kinetic energy for a solid cylinder is KE_rot = (1/2)Iω^2, where I is the moment of inertia and ω is the angular velocity.
  • Step 6: For a solid cylinder, the moment of inertia I = (1/2)mR^2, and the relationship between linear velocity v and angular velocity ω is ω = v/R.
  • Step 7: Substitute ω into the rotational kinetic energy formula to find KE_rot in terms of v: KE_rot = (1/2)(1/2)mR^2(v/R)^2 = (1/4)mv^2.
  • Step 8: Now, add the translational and rotational kinetic energies: KE_total = KE_trans + KE_rot = (1/2)mv^2 + (1/4)mv^2 = (3/4)mv^2.
  • Step 9: To find the ratio of translational kinetic energy to total kinetic energy, use the formula: Ratio = KE_trans / KE_total = ((1/2)mv^2) / ((3/4)mv^2).
  • Step 10: Simplify the ratio: Ratio = (1/2) / (3/4) = (1/2) * (4/3) = 2/3.
  • Step 11: Therefore, the ratio of translational kinetic energy to total kinetic energy is 2:1.
  • Kinetic Energy – Understanding the distinction between translational and rotational kinetic energy in rolling objects.
  • Rolling Motion – The relationship between translational and rotational motion for objects rolling without slipping.
  • Energy Conservation – Applying the principle of conservation of energy to analyze the motion of the cylinder down the incline.
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