A small particle of mass m is pulled to the top of a...

A small particle of mass $m$ is pulled to the top of a frictionless half-cylinder (of radius $R$ ) by a cord that passes over the top of the cylinder, as illustrated in Figure P7.20. (a) If the particle moves at a constant speed, show that $F=m g \cos \theta .$ (Note: If the particle moves at constant speed, the component of its acceleration tangent to the cylinder must be zero at all times.) (b) By directly integrating $W=\int \mathbf{F} \cdot d \mathbf{r},$ find the work done in moving the particle at constant speed from the bottom to the top of the half-cylinder. (FIGURE CAN'T COPY)

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Key Concepts

Work Done by a Force

Work is calculated as the line integral of the force along the path of movement. This concept connects the force acting at each point along a journey to the overall energy transferred or required for displacement. In scenarios where an object is moved along a curved path, correctly setting up and evaluating the line integral is necessary to determine the total work done by the applied force.

Tangential Equilibrium

When an object moves at a constant speed along a curved path, the acceleration along the tangent to the path is zero. This means that the net force acting in the tangential direction must also be zero. Understanding this concept is crucial because it implies that any applied force must exactly balance the component of other forces (like gravity) that would otherwise speed up or slow down the particle along its path.

Force Decomposition

In problems involving motion along curved paths, forces are often decomposed into components that are tangent and normal to the path. The tangential component influences the change in speed while the normal component affects only the direction of motion. This decomposition is essential for analyzing scenarios where the object is constrained to a specific pathway, such as a curved or circular trajectory.

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