A toboggan slides down a hill and has a constant velocity. The angle of the hill is 6.50° with respect to the horizontal. What is the coefficient of kinetic friction between the surface of the hill and the toboggan?
mgsinα=μN=μmgcosα
μ=tanα
To find the coefficient of kinetic friction between the surface of the hill and the toboggan, we need to analyze the forces acting on the toboggan.
First, let's draw a free-body diagram of the forces acting on the toboggan:
1. Gravitational Force (mg): This force acts vertically downwards and has a magnitude equal to the mass of the toboggan (m) multiplied by the acceleration due to gravity (g).
2. Normal Force (N): This force acts perpendicular to the surface of the hill and counteracts the gravitational force. Since the toboggan is on a slope, the normal force is not equal to the weight (mg), but rather the component of the weight that is perpendicular to the surface of the hill. Therefore, the normal force can be calculated as N = mg * cos(θ), where θ is the angle of the hill.
3. Frictional Force (f): This force acts parallel to the surface of the hill and opposes the motion of the toboggan. The magnitude of the frictional force can be calculated as f = μk * N, where μk is the coefficient of kinetic friction, and N is the normal force.
Since the toboggan is sliding down the hill with a constant velocity, we know that the net force acting on the toboggan must be zero. In other words, the gravitational force and the frictional force must balance each other out horizontally.
Equating the horizontal components of the gravitational force and frictional force:
mg * sin(θ) = μk * N
Since N = mg * cos(θ), we can substitute this expression into the equation:
mg * sin(θ) = μk * mg * cos(θ)
The mass (m) and acceleration due to gravity (g) are common to both sides of the equation, so they cancel out:
sin(θ) = μk * cos(θ)
To find μk, divide both sides of the equation by cos(θ):
μk = sin(θ) / cos(θ)
Now, substitute the given angle θ = 6.50° into the equation:
μk = sin(6.50°) / cos(6.50°)
Using a calculator, you can find the value of sin(6.50°) and cos(6.50°), and then calculate the coefficient of kinetic friction (μk).
To find the coefficient of kinetic friction between the surface of the hill and the toboggan, we can make use of the fact that the toboggan has a constant velocity. This means that the net force acting on the toboggan must be zero.
Let's break down the forces acting on the toboggan:
1. Gravity: The force due to gravity can be split into two components - one parallel to the hill and the other perpendicular to it. The component parallel to the hill is given by mg * sinθ, where m is the mass of the toboggan and g is the acceleration due to gravity (9.8 m/s²). The component perpendicular to the hill is given by mg * cosθ, which is canceled out since the hill has no vertical motion.
2. Friction: The frictional force acts in the opposite direction of the toboggan's motion and is given by the formula F_friction = μ * N, where μ is the coefficient of kinetic friction and N is the normal force. The normal force is equal to mg * cosθ, which again cancels out in the horizontal direction.
Since the net force is zero, we have:
F_net = F_friction + F_parallel = 0
μ * N + mg * sinθ = 0
Since N = mg * cosθ, we can substitute it in:
μ * (mg * cosθ) + mg * sinθ = 0
Now, we can simplify the equation:
μ * cosθ + sinθ = 0
Finally, we solve for μ:
μ = -sinθ / cosθ
Plugging in the given angle θ = 6.50°:
μ = -sin(6.50°) / cos(6.50°)
Using a scientific calculator, we find:
μ ≈ -0.113
Therefore, the coefficient of kinetic friction between the surface of the hill and the toboggan is approximately -0.113. Note that the negative sign indicates that the frictional force is acting in the opposite direction of the toboggan's motion.