Objects of masses m1 = 4.00 kg and m2 = 9.00 kg are connected by a light string that passes over a frictionless pulley as in the figure below. The object m1 is held at rest on the floor, and m2 rests on a fixed incline of θ = 41.0°. The objects are released from rest, and m2 slides 1.80 m down the slope of the incline in 4.70 s.
Determine the coefficient of kinetic friction between m2 and the incline.
To determine the coefficient of kinetic friction between m2 and the incline, we need to break down the forces acting on the system and apply Newton's laws of motion.
First, let's identify the forces acting on m2. The weight of m2 (mg) can be split into two components: one parallel to the incline (mg * sinθ) and one perpendicular to the incline (mg * cosθ).
The force of gravity along the incline (mg * sinθ) will be balanced by the component of the force of friction acting up the incline (friction force = μ * normal force), where μ is the coefficient of kinetic friction. The normal force (mg * cosθ) will be balanced by the component of the object's weight perpendicular to the incline.
Since the objects are connected by a light string, the tension in the string will be the same throughout the system.
Now, we can apply Newton's second law of motion to each object. For m1 on the floor, the net force is zero since it is held at rest. Therefore, we don't need to consider m1 in our calculations.
For m2 on the incline, the net force along the incline is given by Fnet = m2 * acceleration. The acceleration can be determined using the given displacement and time: acceleration = (change in velocity) / time = (final velocity - initial velocity) / time.
We are given that m2 slides 1.80 m down the slope of the incline in 4.70 s. This allows us to calculate the final velocity of m2 using the equation v = u + at, where v is the final velocity, u is the initial velocity (which is zero since it starts from rest), a is the acceleration, and t is the time. Rearranging the equation, we have v = at.
Once we have the final velocity, we can calculate the acceleration using the equation a = (v - u) / t.
Now, we have everything we need to calculate the coefficient of kinetic friction. Rearranging the equation for the net force along the incline, we get:
Fnet = m2 * acceleration = mg * sinθ - friction force
Substituting the value of the friction force as μ * (mg * cosθ), we find:
m2 * acceleration = mg * sinθ - μ * (mg * cosθ)
Simplifying the equation, we get:
μ = (m2 * acceleration + mg * sinθ) / (mg * cosθ)
Substituting the values given in the problem, including the mass of m2, the angle θ, the acceleration, and the acceleration due to gravity (g), you can now solve for the coefficient of kinetic friction μ.