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 Figure P4.48. The object m1 is held at rest on the floor, and m2 rests on a fixed incline of θ = 36.0°. The objects are released from rest, and m2 slides 1.40 m down the incline in 3.70 s.

Figure P4.48
(a) Determine the acceleration of each object.

(b) Determine the tension in the string.


(c) Determine the coefficient of kinetic friction between m2 and the incline.

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To solve this problem, we will use Newton's laws of motion and some basic concepts of force and acceleration.

(a) To determine the acceleration of each object, we can start by analyzing the forces acting on them. Let's consider m1 first. Since it is held at rest on the floor, the only force acting on it is its weight, mg, where g is the acceleration due to gravity (9.8 m/s^2). Therefore, the net force acting on m1 is zero, and its acceleration is also zero.

Next, let's consider m2 on the incline. The forces acting on m2 are its weight, mg, directed vertically downward, and the tension in the string acting parallel to the incline, which we'll call T.

The weight of m2 can be resolved into two components: one perpendicular to the incline, mg*cos(θ), and one parallel to the incline, mg*sin(θ), where θ is the angle of the incline (36.0°).

Since m2 is sliding down the incline, the parallel component of its weight (mg*sin(θ)) will accelerate it downward. The net force acting on m2 along the incline is the difference between the parallel component of its weight and the tension in the string. Therefore, we can write the equation:

Net force = m2 * acceleration

(m2 * g * sin(θ)) - T = m2 * acceleration

We know that the acceleration of m2 is given by the distance it slides divided by the time it takes:

acceleration = (change in velocity) / time

In this case, the change in velocity is the distance it slides (1.40 m) divided by the time it takes (3.70 s). So,

acceleration = 1.40 m / 3.70 s

(b) To determine the tension in the string, we can use the equation we derived earlier:

(m2 * g * sin(θ)) - T = m2 * acceleration

We know the values of m2, g, θ, and acceleration. Rearranging the equation, we can solve for T:

T = m2 * g * sin(θ) - m2 * acceleration

(c) To determine the coefficient of kinetic friction between m2 and the incline, we need to consider the frictional force acting on m2. The frictional force can be calculated using the equation:

Frictional force = coefficient of kinetic friction * normal force

The normal force is the component of the weight of m2 perpendicular to the incline, which is given by mg * cos(θ). Therefore, the frictional force can be written as:

Frictional force = coefficient of kinetic friction * mg * cos(θ)

Knowing that the frictional force is also equal to the net force acting on m2 perpendicular to the incline (since there is no acceleration in that direction), we can set up the equation:

Frictional force = m2 * acceleration_perpendicular

Finally, by rearranging this equation and using the known values, we can solve for the coefficient of kinetic friction:

coefficient of kinetic friction = (m2 * acceleration_perpendicular) / (m2 * g * cos(θ))