A child insists on going sledding on a barely snow-covered hill. The child starts
from rest at the top of the 60 m long hill which is inclined at an angle of 30o to the horizontal, and arrives at the bottom 8.0 s later. What is the coefficient of kinetic friction between the hill and the sled?
Assume that the acceleration is constant, such that
60 m = (a/2)t^2
a = 120/(8)^2 = 1.875 m/s^2
Then use Newton's second law in the form
Fnet = m*g sin30 - m*g*cos30*u = m*a
Note that the mass cancels out, which is good since they did not provide a value.
With the value of a that you now know, the friction coefficient u can be calculated.
0.50 g - 0.866 u*g = 1.875 m/s^2
8.49 u = 4.90 -1.875 = 3.02 m/s^2
u = 0.356
To find the coefficient of kinetic friction between the hill and the sled, we can break down the problem into several steps:
Step 1: Determine the acceleration of the sled.
Since the child starts from rest, the initial velocity (v₀) is 0 m/s. The final velocity (v) can be calculated using the equation v = u + at, where "u" is the initial velocity, "a" is the acceleration, and "t" is the time taken. In this case, the final velocity is not given, but we know that the sled arrives at the bottom of the hill with an acceleration equal to the acceleration due to gravity (g). Therefore, we can write the equation as v = u + gt. Solving for "a", we get a = (v - u) / t. Since u = 0, the equation simplifies to a = v / t. Substituting the given values, a = 0 / 8.0 = 0 m/s².
Step 2: Find the component of the gravitational force parallel to the incline.
The force of gravity acting on the sled can be split into two components - the component parallel to the incline and the component perpendicular to the incline. The component parallel to the incline, which is causing the sled to move downhill, can be calculated using the equation F_parallel = m * g * sinθ, where "m" is the mass of the sled, "g" is the acceleration due to gravity, and "θ" is the angle of the incline. Since the mass of the sled is not given, we can ignore it for now.
Step 3: Calculate the net force acting on the sled.
The net force acting on the sled is the difference between the force parallel to the incline and the force of kinetic friction. Since the sled is in motion, we need to consider the force of kinetic friction opposing the motion. The equation for the force of kinetic friction is F_kinetic = μ * N, where "μ" is the coefficient of kinetic friction and "N" is the normal force. The normal force can be calculated as N = m * g * cosθ, where "m" is the mass of the sled, "g" is the acceleration due to gravity, and "θ" is the angle of the incline.
Step 4: Setting up the equation.
Since the sled is moving downhill, the net force in the direction of motion is equal to the force parallel to the incline minus the force of kinetic friction. Therefore, we can write the equation as: F_net = F_parallel - F_kinetic = m * a.
Step 5: Solve for the coefficient of kinetic friction.
Substituting the values we found into the equation, we get (m * g * sinθ) - (μ * m * g * cosθ) = m * a.
Next, we need to cancel out the mass "m" from both sides of the equation:
g * sinθ - μ * g * cosθ = a.
Simplifying further, we get:
g * (sinθ - μ * cosθ) = a.
Finally, isolating the coefficient of kinetic friction, we divide both sides of the equation by g:
μ = (g * (sinθ - a / g)) / cosθ.
Now we can substitute the known values:
μ = (9.8 * (sin 30° - 0 / 9.8)) / cos 30°.
Evaluating the expression gives us:
μ ≈ 0.262.
Therefore, the coefficient of kinetic friction between the hill and the sled is approximately 0.262.