A wheel with radius R, mass m, and rotational inertia I rolls without slipping across the ground. If its translational kinetic energy is E, what is its rotational kinetic energy?
rotation kenetic energy is represented by rotK = 1/2Iw(omega)sqared
To find the rotational kinetic energy of the wheel, we need to use the relationship between rotational kinetic energy and rotational inertia.
The rotational kinetic energy (KE_rot) is given by the formula:
KE_rot = (1/2) * I * w^2
Where I is the moment of inertia (rotational inertia) of the wheel and w is the angular velocity of the wheel.
Since the wheel is rolling without slipping, there is a relationship between the rotational and translational motion. The angular velocity is related to the linear velocity (v) by the equation:
v = R * w
Where R is the radius of the wheel.
We can rewrite the rotational kinetic energy formula using this relationship:
KE_rot = (1/2) * I * (v/R)^2
We are given the radius of the wheel (R), the mass of the wheel (m), and the translational kinetic energy (E). To find the rotational kinetic energy, we need to first calculate the moment of inertia (I).
The moment of inertia for a solid wheel rotating about its central axis is given by the formula:
I = (1/2) * m * R^2
Now we can substitute the moment of inertia into the rotational kinetic energy formula:
KE_rot = (1/2) * (1/2) * m * R^2 * (v/R)^2
Simplifying, we get:
KE_rot = (1/4) * m * v^2
Since we are given the translational kinetic energy (E), which is also given by the formula:
E = (1/2) * m * v^2
We can rewrite the equation for rotational kinetic energy as:
KE_rot = (1/4) * E
Therefore, the rotational kinetic energy is equal to one-fourth of the translational kinetic energy:
KE_rot = (1/4) * E