Understanding the Problem
* Rolling without slipping: This means the point of contact between the disc and the plane is instantaneously at rest.
* Kinetic Energy: The energy of motion. For a rolling object, it has two components:
* Translational Kinetic Energy: Due to the disc's linear motion.
* Rotational Kinetic Energy: Due to the disc's spinning motion.
Formulas
* Translational Kinetic Energy (KE_t): KE_t = (1/2) * m * v^2
* m = mass of the disc
* v = linear velocity of the disc
* Rotational Kinetic Energy (KE_r): KE_r = (1/2) * I * ω^2
* I = moment of inertia of the disc (for a solid disc, I = (1/2) * m * r^2)
* ω = angular velocity of the disc
Relating Linear and Angular Velocity
* For a rolling object without slipping, v = r * ω, where 'r' is the radius of the disc.
Calculations
1. Translational Kinetic Energy:
* KE_t = (1/2) * 2 kg * (4 m/s)^2 = 16 J
2. Moment of Inertia: We need the radius (r) of the disc to calculate I. Let's assume the radius is 'r' meters.
* I = (1/2) * 2 kg * r^2 = r^2 kg m^2
3. Angular Velocity:
* ω = v / r = 4 m/s / r
4. Rotational Kinetic Energy:
* KE_r = (1/2) * r^2 kg m^2 * (4 m/s / r)^2 = 8 J
5. Total Kinetic Energy:
* KE_total = KE_t + KE_r = 16 J + 8 J = 24 J
Therefore, the kinetic energy of the rolling disc is 24 J. Notice that the final answer depends on the radius of the disc.
