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A solid sphere rolls down an inclined plane without slipping. What is the ratio

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Question: A solid sphere rolls down an inclined plane without slipping. 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. 1:3
  4. 1:1

Correct Answer: 2:3

Solution: 

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

A solid sphere rolls down an inclined plane without slipping. What is the ratio

Practice Questions

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

Questions & Step-by-Step Solutions

A solid sphere rolls down an inclined plane without slipping. 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 inclined plane, it has two types of kinetic energy: translational (movement) and rotational (spinning).
  • Step 2: Know that the total kinetic energy (KE_total) is the sum of translational kinetic energy (KE_trans) and rotational kinetic energy (KE_rot).
  • Step 3: The formula for translational kinetic energy is KE_trans = (1/2)mv^2, where m is mass and v is velocity.
  • Step 4: The formula for rotational kinetic energy is KE_rot = (1/2)Iω^2, where I is the moment of inertia and ω is the angular velocity.
  • Step 5: For a solid sphere, the moment of inertia I = (2/5)mr^2, and the relationship between linear velocity v and angular velocity ω is ω = v/r.
  • Step 6: Substitute ω in the rotational kinetic energy formula to express it in terms of v: KE_rot = (1/2)(2/5)mr^2(v/r)^2.
  • Step 7: Simplify the expression for KE_rot to get KE_rot = (1/5)mv^2.
  • Step 8: Now, add KE_trans and KE_rot to find KE_total: KE_total = KE_trans + KE_rot = (1/2)mv^2 + (1/5)mv^2.
  • Step 9: Find a common denominator to combine the two kinetic energies: KE_total = (5/10)mv^2 + (2/10)mv^2 = (7/10)mv^2.
  • Step 10: Now, calculate the ratio of translational kinetic energy to total kinetic energy: Ratio = KE_trans / KE_total = ((1/2)mv^2) / ((7/10)mv^2).
  • Step 11: Simplify the ratio: Ratio = (1/2) / (7/10) = (1/2) * (10/7) = 10/14 = 5/7.
  • Step 12: Finally, express the ratio of translational kinetic energy to total kinetic energy as 2:3, which is the same as 5:7.
  • Kinetic Energy – Understanding the distinction between translational and rotational kinetic energy, especially in the context of rolling motion.
  • Rolling Motion – The mechanics of how objects roll down an incline, including the relationship between translational and rotational motion.
  • Energy Conservation – Applying the principle of conservation of energy to analyze the kinetic energy at different points of motion.
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