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

Problem Statement:

Two blocks of masses m1 = 15 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 when the spring has stretched by a length x = 0.01 m. We consider here that the masses of the pulley and the rope are negligible. Determine the tension of the rope. The stiffness of the spring is K = 1000 N / m.

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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 as well as the reference frames we will use 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 previous figure 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:

From equation (2) we obtain the normal magnitude:

In addition, the friction force magnitude and the spring recovery force are given respectively by:

The second equation is called Hooke’s law; in this case we are using the force magnitude.

After substituting these three expressions in equation (1) we obtain:

Mass 2:

In the following figure we have represented the projections of the weight vector of mass 2 that we will use when projecting Newton’s second law on the axes.

Newton’s second law applied to mass 2 is:

And projecting it on the axes we get:

From equation (4) we obtain the magnitude of the normal force that the plane exerts on block (2). If a friction force would act on this block, we would use the normal magnitude to calculate it.

To calculate the acceleration of the blocks we use equations (1) and (3), which constitute a system of two equations with two unknowns variables (T and a). We sum the two equations to eliminate T:

We isolate the acceleration from the previous equation and after substituting the givens we get:

To calculate the tension we substitute the acceleration in equation (1):

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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