A uniform solid sphere of radius R, mass M, and moment of inertia ܫ= ଶ ହ ଶܴܯ is rolling without slipping along a horizontal surface. Its total kinetic energy is the sum of the energies associated with translation of the center of mass and rotation about the center of mass. Find the fraction of the sphere’s total kinetic energy that is attributable to rotation.

To determine the fraction of a uniform solid sphere's total kinetic energy that is attributable to rotation, we need to analyze both the translational and rotational kinetic energy components. Let's break this down step by step.

Understanding Kinetic Energy Components

The total kinetic energy (KE) of a rolling object is the sum of its translational kinetic energy (TKE) and its rotational kinetic energy (RKE). For a sphere rolling without slipping, these components are defined as follows:

• Translational Kinetic Energy (TKE): This is given by the formula: TKE = (1/2)Mv², where M is the mass of the sphere and v is the velocity of the center of mass.
• Rotational Kinetic Energy (RKE): This is calculated using the formula: RKE = (1/2)Iω², where I is the moment of inertia and ω is the angular velocity.

Moment of Inertia for a Solid Sphere

For a uniform solid sphere, the moment of inertia I about its center of mass is given by:

I = (2/5)MR²

Relating Linear and Angular Velocity

Since the sphere rolls without slipping, there is a relationship between the linear velocity v and the angular velocity ω:

v = Rω

Calculating Kinetic Energies

Now, let's express both kinetic energies in terms of the linear velocity v.

Translational Kinetic Energy

Using the formula for TKE:

TKE = (1/2)Mv²

Rotational Kinetic Energy

Substituting the moment of inertia into the RKE formula:

RKE = (1/2)(2/5)MR²ω²

Now, substituting ω = v/R into the RKE equation:

RKE = (1/2)(2/5)MR²(v/R)²

This simplifies to:

RKE = (1/2)(2/5)M(v²/R²)R² = (1/5)Mv²

Finding the Total Kinetic Energy

The total kinetic energy of the sphere is the sum of TKE and RKE:

Total KE = TKE + RKE = (1/2)Mv² + (1/5)Mv²

To combine these, we need a common denominator:

Total KE = (5/10)Mv² + (2/10)Mv² = (7/10)Mv²

Calculating the Fraction of Kinetic Energy Attributable to Rotation

Now, we can find the fraction of the total kinetic energy that is due to rotation:

Fraction of RKE = RKE / Total KE = [(1/5)Mv²] / [(7/10)Mv²]

Notice that Mv² cancels out:

Fraction of RKE = (1/5) / (7/10) = (1/5) * (10/7) = 2/7

Final Result

Thus, the fraction of the sphere’s total kinetic energy that is attributable to rotation is:

2/7

This means that approximately 28.57% of the total kinetic energy of the rolling sphere is due to its rotation about its center of mass. This analysis highlights the interplay between translational and rotational motion in rolling objects.