Newton's laws - Acceleration of two masses with a pulley and a rope

Problem Statement:

Two blocks of masses m1 = 5 kg and m2 = 30 kg are located respectively on a horizontal plane and on an inclined plane with an angle α = 30º (see figure). Assuming that the coefficient of friction between the first block and the horizontal plane is μ = 0.1, determine the acceleration of the two blocks if we consider that the masses of the pulley and the rope are negligible. Determine the tension of the rope.

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Solution:

When trying to solve an application problem of Newton’s second law and there are several masses involved, you will have to write Newton’s second law for each one of them. For each mass, draw the forces that act on it, choose a reference frame and specify the positive direction of the axes.

The blocks are subject to the normal force as they rest on planes. The friction force will also act on the first block. In addition, the weight acts on the blocks because they are close to the Earth. And finally, on each block will act the tension of the rope.

Be specific with the notation. Use indexes to identify the bodies subject to the different forces. The different forces of this problem are represented in the following figure.

Note that in this figure the magnitude of the tension of the rope is the same for both sides of the pulley. This is always the case if the pulley has a negligible mass.

On the other hand, the magnitude of the acceleration of the blocks is the same since they are joined by the rope.

In the figure above you can see that we have chosen a different orientation for the reference frame axes associated with the two masses. In general, when a mass is on an inclined plane, the x-axis is chosen so that it is aligned with the plane. In this way the vector acceleration of the mass needs to be projected only on this axis and its projection corresponds to the magnitude of the vector.

You can orient the axes differently for each of the masses as long as you are consistent with the orientation and the positive direction of the axes you have chosen.

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Next we are going to apply Newton’s second law to each of the blocks.

Mass 1:

With the projections on the axes we obtain:

The magnitude of the friction force is given by:

From equation (2) we obtain the normal magnitude:

And substituting this norm and that of the friction force in equation (1) we obtain:

Mass 2:

The projections of the weight vector of mass 2 are represented in the following figure.

Newton’s second law applied to mass 2 is given by:

And projecting it on the axes we get:

From equation (4) we obtain the normal magnitude on block 2, which would be useful, for example, if we had to calculate the friction force in the inclined plane.

Equation (3) is the one that interests us to solve the problem.

We can now add the equations (1) and (3) to eliminate the tension:

After isolating the acceleration and substituting the givens we obtain:

And after substituting this value in equation (1) we obtain the tension of the rope:

In the problem we have used g = 10 m/s2

Do not forget to include the units in the results of the problem.

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