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If a rolling object has a radius of R and rolls with a speed v, what is its kine

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Question: If a rolling object has a radius of R and rolls with a speed v, what is its kinetic energy?

Options:

  1. (1/2)mv^2
  2. (1/2)mv^2 + (1/2)Iω^2
  3. (1/2)mv^2 + (1/2)(1/2)mR^2(v/R)^2
  4. None of the above

Correct Answer: (1/2)mv^2 + (1/2)(1/2)mR^2(v/R)^2

Solution: 

The total kinetic energy of a rolling object is the sum of translational and rotational kinetic energy, which simplifies to (1/2)mv^2 + (1/4)mv^2 = (3/4)mv^2.

If a rolling object has a radius of R and rolls with a speed v, what is its kine

Practice Questions

Q1
If a rolling object has a radius of R and rolls with a speed v, what is its kinetic energy?
  1. (1/2)mv^2
  2. (1/2)mv^2 + (1/2)Iω^2
  3. (1/2)mv^2 + (1/2)(1/2)mR^2(v/R)^2
  4. None of the above

Questions & Step-by-Step Solutions

If a rolling object has a radius of R and rolls with a speed v, what is its kinetic energy?
  • Step 1: Understand that a rolling object has two types of motion: it moves forward (translational motion) and it spins (rotational motion).
  • Step 2: The translational kinetic energy (TKE) is calculated using the formula TKE = (1/2)mv^2, where m is the mass of the object and v is its speed.
  • Step 3: The rotational kinetic energy (RKE) is calculated using the formula RKE = (1/2)Iω^2, where I is the moment of inertia and ω is the angular velocity.
  • Step 4: For a solid cylinder or sphere, the moment of inertia I can be expressed as (1/2)mR^2, and the angular velocity ω can be related to the speed v by the formula ω = v/R.
  • Step 5: Substitute ω into the RKE formula: RKE = (1/2)(1/2)mR^2(v/R)^2 = (1/4)mv^2.
  • Step 6: Now, add the translational kinetic energy and the rotational kinetic energy together: Total KE = TKE + RKE = (1/2)mv^2 + (1/4)mv^2.
  • Step 7: Combine the terms: (1/2)mv^2 + (1/4)mv^2 = (3/4)mv^2.
  • Step 8: Therefore, the total kinetic energy of the rolling object is (3/4)mv^2.
  • Kinetic Energy of Rolling Objects – The total kinetic energy of a rolling object is the sum of its translational kinetic energy (1/2 mv^2) and its rotational kinetic energy (1/2 Iω^2), where I is the moment of inertia and ω is the angular velocity.
  • Moment of Inertia – For a solid cylinder or disk, the moment of inertia I is (1/2)mR^2, which is crucial for calculating rotational kinetic energy.
  • Relationship Between Linear and Angular Velocity – For rolling without slipping, the relationship v = Rω must be used to relate linear speed to angular speed.
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