**Problem Statement:**

A mass begins its motion at point A without initial velocity and without friction with the track. Determine in terms of h_{A} what may be the maximum value of the radius of the circular part of the track so that the mass does not fall. Then calculate what is the maximum height it would reach at point C.

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**Solution:**

First we are going to calculate the radius of the circular section of the track using Newton’s second law.

We draw the forces acting on the mass at point B (the weight because it is close to the surface of the Earth and the normal because it is on the track):

We have also represented the axes (tangent and perpendicular to the trajectory) that we will use to make the projection of the vectors of Newton’s second law.

Newton’s second law applied to the mass at point B is given by:

Since the two forces acting on the mass are directed along the axis perpendicular to the trajectory, we only project the vectors of Newton’s second law on this axis:

The magnitude of the normal acceleration appears in the second member of equation (1) since the projection of the acceleration vector on the axis perpendicular to the trajectory is by definition the normal acceleration.

On the other hand, the magnitude of the normal acceleration is given by:

And substituting in equation (1) we get:

When the mass falls and stops being in contact with the track, the normal force is canceled. By imposing this condition in equation (1) we will find what is the minimum speed the mass must have at point B to continue on the track. Using the energy conservation principle we will determine the radius R.

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Once obtained the value of the minimum speed we will determine the radius R.

There is no friction between the track and the mass and the normal force does not produce work because it is perpendicular to the displacement, therefore the mechanical energy is preserved between the points A and B:

The mass is at rest initially, the initial speed is therefore zero. The height of the mass at point B is 2R and the velocity at point B is given by equation (1). By substituting we get:

Finally we isolate R from equation (2):

To determine the maximum height reached by the mass at point C we apply the energy conservation principle between points A and C. The last segment of the track is straight, so the speed will be zero when the mass reaches the maximum height h_{C}.

The energy conservation principle between points A and C is:

This result was expected, the mass reaches the same height it started from because its energy is preserved.

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