A solid spherical ball rolls on table. Ration of its rotational kinetic energy to its total kinetic energy.

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To find the ratio of rotational kinetic energy to total kinetic energy for a rolling spherical ball, we need to understand the different components involved.

The total kinetic energy (KE) of the ball is the sum of the translational kinetic energy and rotational kinetic energy. The rotational kinetic energy (KE_rot) is associated with the spinning motion of the ball, while the translational kinetic energy (KE_trans) is related to the linear motion.

The formula for the translational kinetic energy of an object is given by KE_trans = (1/2) * m * v^2, where m is the mass of the object and v is its linear velocity.

The formula for the rotational kinetic energy of a sphere is given by KE_rot = (1/2) * I * ω^2, where I is the moment of inertia of the sphere and ω is its angular velocity.

For a solid sphere rolling without slipping, the relationship between linear velocity (v) and angular velocity (ω) is given by ω = v / r, where r is the radius of the sphere.

Now, let's substitute the value of ω into the formula for rotational kinetic energy:

KE_rot = (1/2) * I * (v/r)^2
= (1/2) * I * v^2 / r^2

The total kinetic energy (KE_total) is the sum of translational and rotational kinetic energy:

KE_total = KE_trans + KE_rot
= (1/2) * m * v^2 + (1/2) * I * v^2 / r^2

Now we can find the ratio of rotational kinetic energy to total kinetic energy:

Ratio = KE_rot / KE_total
= [(1/2) * I * v^2 / r^2] / [(1/2) * m * v^2 + (1/2) * I * v^2 / r^2]

Simplifying the expression further, we get:

Ratio = I / (m * r^2 + I)

So, to find the ratio of rotational kinetic energy to total kinetic energy for a solid spherical ball rolling on a table, you need to know the mass (m) and radius (r) of the ball, as well as the moment of inertia (I) for a sphere. Once you have these values, simply plug them into the expression: Ratio = I / (m * r^2 + I) to find the desired ratio.