Kinetic Energy of a Rolling Disc

To calculate the kinetic energy of a rolling disc, we need to consider both its translational and rotational motion. The formula for the kinetic energy of a rolling disc is given by:

K = (1/2) * I * ω^2 + (1/2) * M * v^2

Where:
- K is the kinetic energy of the disc
- I is the moment of inertia of the disc
- ω is the angular velocity of the disc
- M is the mass of the disc
- v is the linear velocity of the disc

Calculating the Angular Velocity (ω)
The angular velocity of a rolling disc is related to its linear velocity by the equation:

v = ω * r

Where:
- r is the radius of the disc

Rearranging the equation, we can solve for ω:

ω = v / r

Given that the radius of the disc is 50cm and the linear velocity is 10m/s, we can calculate the angular velocity as follows:

ω = 10m/s / 0.5m = 20 rad/s

Calculating the Moment of Inertia (I)
The moment of inertia of a disc is given by the equation:

I = (1/2) * M * r^2

Given that the mass of the disc is 5kg and the radius is 50cm, we can calculate the moment of inertia as follows:

I = (1/2) * 5kg * (0.5m)^2 = 0.625 kg·m^2

Calculating the Kinetic Energy (K)
Now that we have the moment of inertia and angular velocity, as well as the mass and linear velocity, we can substitute these values into the formula for kinetic energy:

K = (1/2) * I * ω^2 + (1/2) * M * v^2

K = (1/2) * 0.625 kg·m^2 * (20 rad/s)^2 + (1/2) * 5kg * (10m/s)^2

Simplifying the equation, we get:

K = 0.5 * 0.625 kg·m^2 * 400 rad^2/s^2 + 0.5 * 5kg * 100 m^2/s^2

K = 125 J + 250 J

K = 375 J

Therefore, the kinetic energy of the disc is 375 Joules.