Final answer:
To find the coefficient of kinetic friction, apply the work-energy principle: calculate the work done by the force and the change in kinetic energy, and then solve for the friction work which includes the coefficient of kinetic friction in its formula.
Step-by-step explanation:
To find the coefficient of kinetic friction (μ_k) between the block and the rough surface, we can apply the work-energy principle and Newton's second law. The work done by the applied force is equal to the change in kinetic energy plus the work done against friction:
- Work by applied force: W = F * d = 50.0 N * 6.45 m
- Kinetic energy final: KE_f = 0.5 * m * v^2 = 0.5 * 12.0 kg * (5.90 m/s)^2
- Kinetic energy initial: KE_i = 0 (since the block is initially at rest)
Assuming all work done by the force is converted to kinetic energy and work against friction, we can say:
W = KE_f - KE_i + Work by friction
Work by friction is the force of friction times distance, which is μ_k * N * d, where N is the normal force (equal to the weight of the block in this case since the surface is horizontal).
Thus:
- Work by friction: W_fric = μ_k * m * g * d
The equation now becomes:
F * d = 0.5 * m * v^2 + μ_k * m * g * d
By isolating μ_k, we can solve for the coefficient of kinetic friction:
μ_k = (F * d - 0.5 * m * v^2) / (m * g * d)
Plugging in the known values:
μ_k = (50.0 N * 6.45 m - 0.5 * 12.0 kg * (5.90 m/s)^2) / (12.0 kg * 9.8 m/s^2 * 6.45 m)
After calculations, we obtain the value for μ_k. This is the coefficient of kinetic friction we were looking to find.