A pendulum consists of a small object hanging from the...

A pendulum consists of a small object hanging from the ceiling at the end of a string of negligible mass. The string has a length of $0.75 \mathrm{m}$. With the string hanging vertically, the object is given an initial velocity of $2.0 \mathrm{m} / \mathrm{s}$ parallel to the ground and swings upward on a circular arc. Eventually, the object comes to a momentary halt at a point where the string makes an angle $\theta$ with its initial vertical orientation and then swings back downward. Find the angle $\theta$.

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Key Concepts

Conservation of Energy

This principle states that in a closed system, energy is neither created nor destroyed but only transformed from one form to another. In the context of the pendulum, the initial kinetic energy provided to the object is converted into gravitational potential energy as the object ascends.

Gravitational Potential Energy

This is the energy an object possesses due to its position in a gravitational field, commonly expressed as mgh. For the pendulum, as it rises to an angle, its height increases and hence its gravitational potential energy increases, which can be related back to the change in height using the geometry of the pendulum.

Pendulum Motion and Geometry

The analysis of pendulum motion involves understanding its path along a circular arc determined by the string’s length. The vertical displacement of the pendulum bob relative to its lowest point is derived from the length of the string and the cosine of the angle of displacement, forming the basis for relating the potential energy change to the angle.

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