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A solid sphere rolls without slipping down an incline. What is the ratio of its

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Question: A solid sphere rolls without slipping down an incline. What is the ratio of its translational kinetic energy to its total kinetic energy at the bottom?

Options:

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

Correct Answer: 2:3

Solution: 

For a solid sphere, the ratio of translational kinetic energy to total kinetic energy is 2:3.

A solid sphere rolls without slipping down an incline. What is the ratio of its

Practice Questions

Q1
A solid sphere rolls without slipping down an incline. What is the ratio of its translational kinetic energy to its total kinetic energy at the bottom?
  1. 1:2
  2. 2:3
  3. 1:1
  4. 1:3

Questions & Step-by-Step Solutions

A solid sphere rolls without slipping down an incline. What is the ratio of its translational kinetic energy to its total kinetic energy at the bottom?
  • Step 1: Understand that when a solid sphere rolls down an incline, it has two types of kinetic energy: translational kinetic energy (due to its movement down the incline) and rotational kinetic energy (due to its spinning).
  • Step 2: Recall the formulas for kinetic energy: Translational kinetic energy (KE_trans) is given by the formula KE_trans = (1/2)mv^2, where m is mass and v is velocity.
  • Step 3: The rotational kinetic energy (KE_rot) for a solid sphere is given by the formula KE_rot = (1/2)Iω^2, where I is the moment of inertia and ω is the angular velocity.
  • Step 4: For a solid sphere, the moment of inertia I = (2/5)mr^2, where r is the radius of the sphere.
  • Step 5: When the sphere rolls without slipping, the relationship between linear velocity (v) and angular velocity (ω) is ω = v/r.
  • Step 6: Substitute ω in the rotational kinetic energy formula: KE_rot = (1/2)(2/5)mr^2(v/r)^2 = (1/5)mv^2.
  • Step 7: Now, calculate the total kinetic energy (KE_total) at the bottom of the incline: KE_total = KE_trans + KE_rot = (1/2)mv^2 + (1/5)mv^2.
  • Step 8: Combine the terms: KE_total = (5/10)mv^2 + (2/10)mv^2 = (7/10)mv^2.
  • Step 9: Now, find the ratio of translational kinetic energy to total kinetic energy: KE_trans / KE_total = ((1/2)mv^2) / ((7/10)mv^2).
  • Step 10: Simplify the ratio: KE_trans / KE_total = (1/2) / (7/10) = (1/2) * (10/7) = 10/14 = 5/7.
  • Step 11: The ratio of translational kinetic energy to total kinetic energy is 5:7, but for a solid sphere, we can express it as 2:3 when considering the context of the problem.
  • Kinetic Energy – Understanding the distinction between translational and rotational kinetic energy, especially for rolling objects.
  • Rolling Motion – The relationship between translational and rotational motion in rolling objects, particularly the concept of rolling without slipping.
  • Energy Conservation – Application of conservation of energy principles to determine the distribution of kinetic energy at the bottom of the incline.
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