A hockey puck is sliding down an inclined plane with angle θ = 12.4° as shown in the figure below. If the puck is moving with a constant speed, what is the coefficient of kinetic friction between the puck and the plane?

5.43

To determine the coefficient of kinetic friction between the puck and the plane, we can use the concept of force balance. In this case, the force of gravity acting down the inclined plane can be resolved into two components: one parallel to the plane and the other perpendicular to the plane.

Let's break down the forces acting on the hockey puck:
1. Gravity force (mg): This force acts vertically downwards and can be split into two components: one parallel to the inclined plane and one perpendicular to it.
- The component parallel to the plane is defined as F_parallel = mg * sin(θ), where m is the mass of the puck and θ is the angle of the inclined plane (12.4°).
- The component perpendicular to the plane is defined as F_perpendicular = mg * cos(θ).

2. Normal force (N): This force is exerted by the inclined plane and acts perpendicular to it.

3. Friction force (f): This force acts parallel to the inclined plane and opposes the motion of the puck. Since the puck is moving with constant speed, the friction force is equal in magnitude and opposite in direction to the parallel component of the weight force.

Now, according to the concept of force balance in the direction perpendicular to the plane, the perpendicular component of the weight force (F_perpendicular) is balanced by the normal force (N). Therefore, N = F_perpendicular = mg * cos(θ).

Next, in the direction parallel to the plane, the sum of the forces must be zero since the puck is moving at a constant speed. The forces acting in this direction are the parallel component of the weight force (F_parallel) and the friction force (f). Hence, F_parallel - f = 0.

Substituting the expressions for F_parallel and N from above, we have mg * sin(θ) - f = 0. Solving for f, we get f = mg * sin(θ).

Since the puck is moving with a constant speed, the friction force is equal to the force of gravity. Hence, f = mg * sin(θ).

Now, we can use the formula for the friction force to find the coefficient of kinetic friction (μ) using the equation f = μ * N.

Since N = mg * cos(θ), we can substitute it into the equation and solve for μ:

μ * (mg * cos(θ)) = mg * sin(θ)

Canceling out the mass (m) and simplifying the equation, we get:

μ * cos(θ) = sin(θ)

Finally, we can solve for the coefficient of kinetic friction (μ):

μ = sin(θ) / cos(θ)

Plugging in the angle θ = 12.4° into the equation, we can calculate the coefficient of kinetic friction.