Question

A circular disc of mass \(\mathrm{m}\) and radius \(\mathrm{r}\) is rolling on a smooth horizontal surface with a constant speed v. Its kinetic energy is \(\\{\mathrm{A}\\}(1 / 4) \mathrm{mv}^{2}\) \(\\{\mathrm{B}\\}(1 / 2) \mathrm{mv}^{2}\) \(\\{\mathrm{C}\\}(3 / 4) \mathrm{mv}^{2}\) \(\\{\mathrm{D}\\} \mathrm{mv}^{2}\)

Short answer

The total kinetic energy of the circular disc is \(\frac{3}{4}mv^2\). Hence, the correct answer is \(\boxed{\text{C} (3 / 4) \mathrm{mv}^{2}}\).

Step-by-step solution

Identify the given variables.

In this problem, we are given the following variables: 1. Mass of the circular disc (m) 2. Radius of the circular disc (r) 3. Constant speed of the circular disc (v)

Calculate the translational kinetic energy.

The translational kinetic energy of the disc, moving with a constant speed v, is given by the formula: Translational Kinetic Energy = \( \frac{1}{2} * m * v^2 \)

Calculate the rotational kinetic energy.

For a rolling object, the moment of inertia (I) for a circular disc about its center of mass is given by the formula: Moment of Inertia (I) = \( \frac{1}{2} * m * r^2 \) The rotational kinetic energy is given by the formula: Rotational Kinetic Energy = \( \frac{1}{2} * I * ω^2 \) Since we are given the constant speed (v) and we know that the object is rolling without slipping, we can find the angular velocity (ω) using the formula: \( ω = \frac{v}{r} \) Now we plug the values into the rotational kinetic energy formula: Rotational Kinetic Energy = \( \frac{1}{2} * (\frac{1}{2} * m * r^2) * (\frac{v}{r})^2 \) Simplify the equation: Rotational Kinetic Energy = \( \frac{1}{4} * m * v^2 \)

Calculate the total kinetic energy.

Now we can find the total kinetic energy by summing both the translational and rotational kinetic energies: Total Kinetic Energy = Translational Kinetic Energy + Rotational Kinetic Energy Total Kinetic Energy = \( (\frac{1}{2} * m * v^2) + (\frac{1}{4} * m * v^2) \) Total Kinetic Energy = \( (\frac{3}{4} * m * v^2) \)

Find the correct answer choice.

Now we have the total kinetic energy of the circular disc as \( (\frac{3}{4} * m * v^2) \). Looking at the given options, the correct answer is: \(\boxed{\text{C} (3 / 4) \mathrm{mv}^{2}}\)