A uniform cord of length 25 cm and mass 15 g is initially stuck to a ceiling. Later, it hangs

step 1 Well, we have length of chord L is equal to 25 centimeters and its mass is equal to 15 grams. We

A uniform cord of length 25 cm and mass 15 g is initially...

A uniform cord of length $25 \mathrm{~cm}$ and mass $15 \mathrm{~g}$ is initially stuck to a ceiling. Later, it hangs vertically from the ceiling with only one end still stuck. What is the change in the gravitational potential energy of the cord with this change in orientation? (Hint: Consider a differential slice of the cord and then use integral calculus.)

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Key Concepts

Gravitational Potential Energy

Gravitational potential energy is the energy an object has due to its position in a gravitational field. For extended objects, the change in gravitational potential energy is determined by considering the work done against gravity when moving each part of the object, which can be summed (or integrated) over its entire mass distribution.

Integration over a Continuous Mass Distribution

When an object’s mass is continuously distributed (as in a cord), you can calculate physical quantities like gravitational potential energy by dividing the object into infinitesimally small mass elements and integrating over the length of the object. This approach uses differential calculus to account for the contribution of each slice of mass at different positions.

Uniform Mass Distribution

A uniform mass distribution means that the object’s mass is evenly spread over its length, area, or volume. In calculations, this allows one to define a constant mass per unit length (or density), simplifying the integration process, especially when determining quantities like the center of mass or gravitational potential energy.

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