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 = 39.0°. The objects are released from rest, and m2 slides 1.80 m down the slope of the incline in 4.50 s.
(a) Determine the acceleration of each object. (Enter the magnitude only.)
(b) Determine the tension in the string. (Enter the magnitude only.)
(c) Determine the coefficient of kinetic friction between m2 and the incline
To solve this problem, we can use Newton's second law of motion, F = ma, and apply it to each object separately. Let's break down the problem into parts.
(a) Determine the acceleration of each object:
For object m1: Since it is held at rest on the floor, the net force acting on it is the tension in the string. Therefore, we can write:
T = m1 * a (equation 1)
For object m2: The forces acting on it are its weight m2g, the tension T, and the friction force. The friction force can be represented as μk * N, where μk is the coefficient of kinetic friction and N is the normal force. The normal force N can be found as N = m2 * g * cos(θ), where θ is the angle of the incline.
Therefore, the net force acting on m2 can be written as:
m2 * g * sin(θ) - μk * m2 * g * cos(θ) = m2 * a (equation 2)
Now, we can substitute the values into the equations:
From equation 1, we have:
T = m1 * a
And from equation 2, we have:
m2 * g * sin(θ) - μk * m2 * g * cos(θ) = m2 * a
Now, we need to solve these two equations simultaneously to find the values of T and a.
(b) Determine the tension in the string:
We can solve equation 1 for T:
T = m1 * a
(c) Determine the coefficient of kinetic friction between m2 and the incline:
We can solve equation 2 for μk:
μk = (m2 * g * sin(θ) - m2 * a) / (m2 * g * cos(θ))
Now, we have the equations necessary to find the acceleration of each object, the tension in the string, and the coefficient of kinetic friction between m2 and the incline.
To find the numerical values, you will need to substitute the given values for m1, m2, θ, and plug them into the above equations. Once you have the values, you can calculate the acceleration, tension, and coefficient of kinetic friction. Remember to check the units and use appropriate values for g (acceleration due to gravity, typically 9.8 m/s²).