A toboggan slides down a hill and has a constant velocity. The angle of the hill is 6.70° with respect to the horizontal. What is the coefficient of kinetic friction between the surface of the hill and the toboggan?
To find the coefficient of kinetic friction between the surface of the hill and the toboggan, we can use the relationship between the angle of inclination and the coefficient of friction.
The force of kinetic friction (Fk) acting on the toboggan can be calculated using the following formula:
Fk = μk * N
where μk is the coefficient of kinetic friction and N is the normal force acting on the toboggan.
Since the toboggan is moving with constant velocity, we can assume that the net force acting on it is zero. Therefore, the force of gravity (mg) acting downwards must be balanced by the force of friction (Fk) acting upwards along the hill.
Now, let's break down the forces acting on the toboggan:
1. The force of gravity (mg) acting straight downwards can be resolved into two components: one along the hill and one perpendicular to it.
- The component acting along the hill (mg sin θ) helps the toboggan slide down the hill.
- The component perpendicular to the hill (mg cos θ) contributes to the normal force acting on the toboggan.
2. The force of friction (Fk) acts in the opposite direction to the motion, up the hill.
Since the toboggan is moving with constant velocity, the force of gravity along the hill (mg sin θ) must be balanced by the force of kinetic friction (Fk). Therefore, we can write the equation:
mg sin θ = μk * N
Now, let's consider the normal force (N) acting on the toboggan. Since the toboggan is on an incline, the normal force is equal to the perpendicular component of the force of gravity (mg cos θ):
N = mg cos θ
Substituting this value of N into the equation above, we get:
mg sin θ = μk * (mg cos θ)
Simplifying the equation, we find:
μk = tan θ
Finally, we can substitute the given angle (θ = 6.70°) into the equation to find the coefficient of kinetic friction:
μk = tan(6.70°)
Using a calculator, we find that the coefficient of kinetic friction (μk) is approximately 0.117.