Final answer:
Using the work-energy principle, the initial and final kinetic energy can be calculated, and with the work done by gravity and friction, we can solve for the coefficient of kinetic friction by equating the total work to the change in kinetic energy and rearranging the equation.
Step-by-step explanation:
To solve for the coefficient of kinetic friction, we must use the work-energy principle, which states that the work done by all forces acting on an object is equal to the change in its kinetic energy. The forces doing work on the mass m are gravity, friction, and the normal force, with only gravity and friction doing non-zero work since the normal force is perpendicular to the motion of the mass.
First, we find the initial kinetic energy (KE_initial) using KE_initial = (1/2) * m * v^2, where m is the mass of the object and v is its initial speed. The final kinetic energy (KE_final) after sliding up the slope is zero because the question indicates that the mass stops. The work done by gravity (W_gravity) is the component of its weight along the incline, which is m * g * sin(θ) * d, where g is the acceleration due to gravity, θ is the angle of the incline, and d is the distance slid. The work done by the frictional force (W_friction) is -f_k * d, where f_k = μ_k * N, with μ_k being the coefficient of kinetic friction and N as the normal force, which is m * g * cos(θ).
Setting up the equation forthe work-energy principle, we have:
KE_initial + W_gravity + W_friction = KE_final
(1/2) * m * v^2 + m * g * sin(θ) * d - μ_k * m * g * cos(θ) * d = 0
After rearranging and solving for μ_k, we find the answer for the coefficient of kinetic friction.