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A solid sphere of mass m and radius r rolls without slipping down an inclined pl

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Question: A solid sphere of mass m and radius r rolls without slipping down an inclined plane of angle θ. What is the acceleration of the center of mass of the sphere?

Options:

  1. g sin(θ)
  2. g sin(θ)/2
  3. g sin(θ)/3
  4. g sin(θ)/4

Correct Answer: g sin(θ)

Solution: 

The acceleration of the center of mass of a rolling object is given by a = g sin(θ) / (1 + k^2/r^2). For a solid sphere, k^2/r^2 = 2/5, thus a = g sin(θ) / (1 + 2/5) = g sin(θ) / (7/5) = (5/7)g sin(θ).

A solid sphere of mass m and radius r rolls without slipping down an inclined pl

Practice Questions

Q1
A solid sphere of mass m and radius r rolls without slipping down an inclined plane of angle θ. What is the acceleration of the center of mass of the sphere?
  1. g sin(θ)
  2. g sin(θ)/2
  3. g sin(θ)/3
  4. g sin(θ)/4

Questions & Step-by-Step Solutions

A solid sphere of mass m and radius r rolls without slipping down an inclined plane of angle θ. What is the acceleration of the center of mass of the sphere?
  • Step 1: Identify the forces acting on the sphere. The main force is gravity, which can be broken down into two components: one parallel to the incline (g sin(θ)) and one perpendicular to the incline (g cos(θ)).
  • Step 2: Understand that the sphere rolls without slipping. This means that as it rolls down, it also rotates. The acceleration of the center of mass (a) is related to the linear motion and the rotational motion.
  • Step 3: Use the formula for the acceleration of a rolling object: a = g sin(θ) / (1 + k^2/r^2), where k is the radius of gyration and r is the radius of the sphere.
  • Step 4: For a solid sphere, the value of k^2/r^2 is 2/5. Substitute this value into the formula: a = g sin(θ) / (1 + 2/5).
  • Step 5: Simplify the expression. The denominator becomes 1 + 2/5 = 7/5. So, a = g sin(θ) / (7/5).
  • Step 6: To simplify further, multiply by the reciprocal: a = (5/7) g sin(θ). This gives the final expression for the acceleration of the center of mass of the sphere.
  • Rolling Motion – Understanding the dynamics of rolling objects, including the relationship between translational and rotational motion.
  • Moment of Inertia – Applying the concept of moment of inertia (k^2) for different shapes, specifically for a solid sphere.
  • Inclined Plane Dynamics – Analyzing forces acting on an object on an inclined plane and how they affect acceleration.
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