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 translation and rotation, we need to calculate the total kinetic energy of the body first.
The total kinetic energy (K) of the rolling body is the sum of the kinetic energy of its center of mass motion (K_trans) and the kinetic energy of rotation about its center of mass (K_rot).
1. Kinetic Energy of center of mass motion (K_trans):
The kinetic energy of center of mass motion is given by the formula:
K_trans = (1/2)Mv^2
where M is the mass of the body and v is the velocity of the center of mass.
2. Kinetic Energy of rotation about the center of mass (K_rot):
The kinetic energy of rotation is given by the formula:
K_rot = (1/2)Iω^2
where I is the moment of inertia of the body and ω is the angular velocity of rotation.
Given that I = BMR^2, we can substitute it into the formula for K_rot:
K_rot = (1/2)BMR^2ω^2
Now, to find the fractions of the body's kinetic energy associated with translation and rotation, we divide each contribution by the total.
Fraction of kinetic energy associated with translation (f_trans):
f_trans = K_trans / K
f_trans = [(1/2)Mv^2] / [K_trans + K_rot]
Fraction of kinetic energy associated with rotation (f_rot):
f_rot = K_rot / K
f_rot = [(1/2)BMR^2ω^2] / [K_trans + K_rot]
Notice that the fractions depend on B as given. By plugging in the values of M, R, v, and ω, you can calculate the fractions associated with translation and rotation.