Masses 19 kg and 9 kg are connected by a light string that passes over a frictionless pulley as shown in the figure ( The figure shows the 19 kg block on the table connected by a string through a pulley which is connected to the 5 kg block hanging off the table). The acceleration due to gravity is 9.8m/s. If the 19 kg mass, initially held at rest on the table, is released and moves 1.2 m in 1.4s, determine the coefficient of kinetic friction between it and the table.

To determine the coefficient of kinetic friction between the 19 kg mass and the table, we need to use Newton's second law of motion and apply it to the system.

First, let's analyze the forces acting on the 19 kg mass. There are two main forces: the gravitational force (mg) and the frictional force (f_k).

The gravitational force is given by:
F_gravity = mg = 19 kg * 9.8 m/s^2 = 186.2 N

The frictional force can be determined using the equation for kinetic friction:
f_k = μ_k * N

Here, N is the normal force, which is equal to the weight or gravitational force acting on the 19 kg mass. Therefore, N = 186.2 N.

Now, we can write the equation for the acceleration of the system:
m * a = F_net = F_gravity - f_k,

where m is the total mass of the system (19 kg + 9 kg = 28 kg), and a is the acceleration.

Since the masses are connected by a light string over a frictionless pulley, the acceleration of both masses will be the same. Let's assume it to be a (positive value).

For the 9 kg mass, we have:
m * a = f_k

Now, we can solve the equations simultaneously. Let's start with the equation for the 19 kg mass:

19 kg * a = 186.2 N - μ_k * 186.2 N

Simplifying, we get:
a = (186.2 N) / (19 kg + 9 kg) - (μ_k * 186.2 N) / 19 kg

Next, we'll use the given data: the 19 kg mass moves 1.2 m in 1.4 s.

The equation for motion of an object undergoing constant acceleration is:
s = ut + (1/2)at^2

Here, s is the distance traveled, u is the initial velocity (which is zero since the mass is released from rest), t is the time, and a is the acceleration.

Substituting the given values into the equation:
1.2 m = 0 + (1/2) * a * (1.4 s)^2

Simplifying, we get:
1.2 m = 0.98 * a m/s^2

Now, we can equate the two expressions for acceleration, as they are equal:
(1.2 m) / 0.98 s^2 = (186.2 N) / (19 kg + 9 kg) - (μ_k * 186.2 N) / 19 kg

Solving for the coefficient of kinetic friction (μ_k), we have:
(μ_k * 186.2 N) / 19 kg = (186.2 / (19 + 9)) / (1.2 / 0.98)

Dividing both sides by (186.2 N / 19 kg), we find:
μ_k = (186.2 / (19 + 9)) / (1.2 / 0.98)

Evaluating this expression, we can find the coefficient of kinetic friction between the 19 kg mass and the table.

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