Newton's laws - The skateboarder

Problem Statement:

Draw the forces acting on the skateboarder of mass m that slides on the cylindrical surface represented in the figure. For simplicity we assume that there is no friction. Calculate the components of the skater acceleration based on the angle that he has ascended.

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Solution:

As we consider that there is no friction, the only forces that act on the skateboarder are its weight and the reaction of the surface. Since we want to calculate the components of the skateboarder’s acceleration, we will orient the Cartesian axes in the directions tangent and perpendicular to the trajectory. To simplify the calculation of the projections we consider the skateboarder as a punctual mass. We call θ the angle that he has ascended.

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Newton’s second law applied to the skateboarder is:

After doing the projection on the axes tangent and perpendicular to the trajectory we get:

Observe in equation (2) that the normal force magnitude can not be canceled out by the weight component in the normal direction when skateboarder is moving. This would occur only when the velocity of the skater is zero, at the ends of its trajectory. At this point the normal acceleration would be zero too.

From the previous equations we can deduce:

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