A round body of mass M, radius R, and moment of inertia I= BMR^2 is rolling without slipping. Its kinetic energy can be decomposed into the energy of its center of mass motion (translation) and the enrgy of rotation about its center of mass. The fractions of the body's kinetc energy associated with these contributions depend only on B, Find these fractions...
To find the fractions of the body's kinetic energy associated with the contributions of translation and rotation, we need to understand the concept of kinetic energy and how it is distributed in a rolling body.
Kinetic energy is the energy possessed by an object due to its motion. In the case of a rolling body, such as the one described, there are two components of motion: translation (movement of the center of mass) and rotation (spin about the center of mass).
Let's denote the total kinetic energy of the rolling body as KE(total). We can then express this as the sum of the kinetic energy of translation (KE(trans)) and the kinetic energy of rotation (KE(rot)):
KE(total) = KE(trans) + KE(rot)
The kinetic energy of translation (KE(trans)) is given by the formula:
KE(trans) = (1/2) M v^2
where M is the mass of the body and v is the linear velocity of the center of mass. Since the body is rolling without slipping, the linear velocity (v) can be related to the angular velocity (ω) and the radius of the body (R) using the equation v = Rω. Substituting this relationship into the equation for KE(trans), we get:
KE(trans) = (1/2) M (Rω)^2 = (1/2) M R^2 ω^2
The kinetic energy of rotation (KE(rot)) is given by the formula:
KE(rot) = (1/2) I ω^2
where I is the moment of inertia of the body and ω is the angular velocity. In the given problem, the moment of inertia is given as I = BMR^2. Substituting this relationship into the equation for KE(rot), we get:
KE(rot) = (1/2) BMR^2 ω^2
Now, substituting the expressions for KE(trans) and KE(rot) into the equation for KE(total), we have:
KE(total) = (1/2) M R^2 ω^2 + (1/2) BMR^2 ω^2
= [(1/2) M + (1/2) BM] R^2 ω^2
We can see that the fraction of the body's kinetic energy associated with translational motion is:
Fraction of KE(trans) = (1/2) M / [(1/2) M + (1/2) BM]
= M / (M + BM)
Similarly, the fraction of the body's kinetic energy associated with rotational motion is:
Fraction of KE(rot) = (1/2) BM / [(1/2) M + (1/2) BM]
= BM / (M + BM)
Therefore, the fractions of the body's kinetic energy associated with translation and rotation are M / (M + BM) and BM / (M + BM), respectively. These fractions depend only on the value of B, as stated in the problem.