A 55 kg skateboarder wants to just make it to the upper...

$A \quad 55 \quad \mathrm{kg}$ skateboarder wants to just make it to the upper edge of a "half-pipe" with a radius of $3.0 \mathrm{m},$ as shown in Figure $\mathrm{P} 10.74$ What speed does he need at the bottom if he will coast all the way up? The skateboarder isn't a simple particle: Assume that his mass in a deep crouch is concentrated $0.75 \mathrm{m}$ from the half-pipe. If he remains in that position all the way up, what initial speed does he need to reach the upper edge?

Key Concepts

Conservation of Energy

This principle states that in a system where only conservative forces (like gravity) are doing work, the total mechanical energy (the sum of kinetic and potential energy) remains constant. In the context of the problem, this means that the kinetic energy the skateboarder has at the bottom is completely converted into gravitational potential energy as he ascends the half?pipe, allowing us to set up an energy balance to determine the required initial speed.

Gravitational Potential Energy

Gravitational potential energy is the energy stored in an object due to its position in a gravitational field, calculated as the product of an object's mass, the gravitational acceleration, and its height above a reference point. In this problem, the change in potential energy is determined by the vertical height the skateboarder's center of mass is raised as he reaches the upper edge of the half?pipe.

Kinetic Energy

Kinetic energy is the energy of motion, and for a translational motion of a body it is given by one half the mass times the square of its speed. For the simple particle analysis of the skateboarder, his entire energy is translational. However, if we recognize that he is an extended body with a distributed mass, additional considerations of kinetic energy related to rotational motion (if he is rotating about a point) can come into play.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion, and it depends on how the object's mass is distributed relative to the axis of rotation. In problems such as this, where the skateboarder is not a point mass but has a mass distribution (with mass concentrated at a distance from the curvature’s center when in a crouched position), the moment of inertia must be considered to correctly account for the rotational contribution to the energy, altering the energy budget compared to that of a simple particle.

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