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If a rolling object has a mass m and radius r, what is the expression for its to

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Question: If a rolling object has a mass m and radius r, what is the expression for its total kinetic energy?

Options:

  1. (1/2)mv^2
  2. (1/2)mv^2 + (1/2)Iω^2
  3. (1/2)mv^2 + (1/2)mr^2ω^2
  4. (1/2)mv^2 + (1/2)(2/5)mr^2(ω^2)

Correct Answer: (1/2)mv^2 + (1/2)(2/5)mr^2(ω^2)

Solution: 

The total kinetic energy of a rolling object is the sum of translational and rotational kinetic energy, which can be expressed as (1/2)mv^2 + (1/2)(2/5)mr^2(ω^2).

If a rolling object has a mass m and radius r, what is the expression for its to

Practice Questions

Q1
If a rolling object has a mass m and radius r, what is the expression for its total kinetic energy?
  1. (1/2)mv^2
  2. (1/2)mv^2 + (1/2)Iω^2
  3. (1/2)mv^2 + (1/2)mr^2ω^2
  4. (1/2)mv^2 + (1/2)(2/5)mr^2(ω^2)

Questions & Step-by-Step Solutions

If a rolling object has a mass m and radius r, what is the expression for its total 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 given by the formula (1/2)mv^2, where m is the mass and v is the linear velocity of the object.
  • Step 3: The rotational kinetic energy (RKE) is given by the formula (1/2)Iω^2, where I is the moment of inertia and ω is the angular velocity.
  • Step 4: For a solid sphere or cylinder, the moment of inertia I can be expressed as (2/5)mr^2, where r is the radius of the object.
  • Step 5: Substitute the moment of inertia into the rotational kinetic energy formula: RKE = (1/2)(2/5)mr^2(ω^2).
  • Step 6: Combine the translational and rotational kinetic energy to find the total kinetic energy: Total KE = TKE + RKE.
  • Step 7: Write the final expression for total kinetic energy: Total KE = (1/2)mv^2 + (1/2)(2/5)mr^2(ω^2).
  • Translational Kinetic Energy – The energy due to the motion of the center of mass of the object, calculated as (1/2)mv^2.
  • Rotational Kinetic Energy – The energy due to the rotation of the object around its axis, calculated as (1/2)Iω^2, where I is the moment of inertia.
  • Moment of Inertia – A measure of an object's resistance to changes in its rotation, which for a solid sphere is (2/5)mr^2.
  • Rolling Motion – The combination of translational and rotational motion that occurs when an object rolls without slipping.
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