Newton's laws - Inclined plane and wedge

Problem Statement:

A block of mass m = 1 kg is located on a wedge of mass M that descends along an inclined plane with an angle α = 30º without friction (see figure). Calculate the magnitude of the reaction of the wedge on the block assuming that the block does not slide with respect to the wedge.

Solve the problem with respect to a reference frame located in the wedge.

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Solution:

Firstly, we will calculate the acceleration of the wedge-block system with respect to a reference frame at rest on the ground. The system is subject to the normal force as it rests on the plane. It is also subject to the weight if we consider that it is close to the Earth. These forces are shown in the figure below.

Where PT is the weight of the system of the two masses and N the normal that the inclined plane exerts on it.

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Next we are going to apply Newton’s second law to the system of the two bodies:

The projections of the weight vector on the Cartesian axes of the reference frame are represented on the following figure:

The projection of Newton’s second law for the system on the axes is:

After substituting the weight in equation (1) and isolating the acceleration we obtain:

Knowing the acceleration value will allow us to calculate the value of the inertial force acting on the block when the reference frame moves along with the wedge. Remember that the inertial force magnitude is always proportional to the mass on which it acts and to the acceleration of the observer.


Next we will represent the forces acting on the block of mass m with respect to a reference system O’ located in the wedge. The observer O’ is not inertial because it has acceleration.

As you can see in the figure, an inertial force acts on the block because we are observing its motion from a non-inertial reference system.

The inertial force magnitude (using the acceleration value calculated above) is:

The block is also subject to a static friction; if it were not the case, it would not move together with the wedge. In addition, as it rests on the wedge, a normal force is exerted by it and if we consider that it is close to the Earth, the weight will also act on it.

Newton’s second law applied to the block with respect to O’ is:

The projection on the axes are:

The projections of the inertial force on the axes are shown in the figure below. As you can see in the previous figure, the projections of the other forces coincide with their respective magnitudes.

And finally we can isolate the magnitude of the normal N21 from equation (4):

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