The sled is stuck in the snow. A child pulls on the rope and finds that the sled just barely begins to move when he pulls with a force of 24 N, with the rope at an angle of θ = 33° with respect to the horizontal. The mass of the sled is 14 kg.
If the child continues to pull on the sled and it has an acceleration of 0.31 m/s2, find the coefficient of kinetic friction between the sled and the snow.
ma= Fcos θ –μ(mg-Fsin θ)
μ = (Fcos θ -ma)/ (mg-Fsin θ)
^^ That is incorrect
This is correct solution
To find the coefficient of kinetic friction, we first need to determine the net force acting on the sled. Since the sled is moving with an acceleration, there must be a net force acting on it. The net force can be calculated using Newton's second law of motion:
F_net = ma
Where:
F_net is the net force (unknown)
m is the mass of the sled (14 kg)
a is the acceleration of the sled (0.31 m/s^2)
Substituting the known values into the equation, we get:
F_net = (14 kg)(0.31 m/s^2)
F_net = 4.34 N
Next, we can analyze the forces acting on the sled. The child is pulling the sled with a force of 24 N at an angle of 33° with respect to the horizontal. We can resolve this force into its horizontal and vertical components:
F_horizontal = F_child * cos(θ)
F_vertical = F_child * sin(θ)
Substituting the values, we get:
F_horizontal = (24 N) * cos(33°)
F_horizontal = 20 N
F_vertical = (24 N) * sin(33°)
F_vertical = 13 N
Since the sled is not moving vertically, the vertical forces are balanced. This means that the normal force exerted by the snow upward must equal the vertical component of the child's pull:
F_normal = F_vertical = 13 N
Now, let's consider the horizontal forces. The only horizontal force acting on the sled is the force of kinetic friction (F_friction). The force of kinetic friction can be calculated using:
F_friction = μ_k * F_normal
Where:
μ_k is the coefficient of kinetic friction (unknown)
Substituting the known values, we get:
F_friction = μ_k * 13 N
Finally, we can equate the net force to the force of friction and solve for the coefficient of kinetic friction:
F_friction = F_net
μ_k * 13 N = 4.34 N
Simplifying the equation, we get:
μ_k = 4.34 N / 13 N
μ_k ≈ 0.334
Therefore, the coefficient of kinetic friction between the sled and the snow is approximately 0.334.