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Gravitational potential energy is the energy an object possesses because of its position in a gravitational field. When the cord is initially horizontal on the ceiling, its gravitational potential energy is zero because it is at height zero. When it hangs vertically, the cord's segments are at varying heights, accumulating gravitational potential energy as a result.

To calculate the change in gravitational potential energy, we determine how much energy each small segment gains when the cord is hanging. By summing the energy of all these segments, we find the total change.

• The differential segment at height \(y\) from the ceiling contributes \(dU = dm \cdot g \cdot y\) to the gravitational potential energy, where \(g\) is the acceleration due to gravity.
• This energy integrates over the length of the cord from its full height \(25\ \text{cm}\) to zero.

This integration provides the total change in gravitational potential energy when the orientation of the cord changes.

Integral calculus is vital in understanding and calculating changes in gravitational potential energy, especially when dealing with varying heights or distributions, like in this cord problem.

The potential energy of a uniform cord hanging from the ceiling can be calculated by integrating over its length. By dividing the cord into infinitesimally small pieces, we individually calculate the potential energy for each piece and then use integration to sum them.

• The formula \(U = \int_0^{0.25} 0.06 \cdot g \cdot y \, dy\) integrates the function of potential energy with respect to height \(y\), covering the entire length from ceiling to floor.
• Integration effectively adds up all the potential energies of these segments, providing the total energy change when the cord shifts from horizontal to vertical.

This approach highlights how integral calculus is a powerful tool in physics, allowing us to handle complex variable scenarios with precision.

Linear mass density is a critical concept in solving this problem. It represents the mass per unit length of an object, such as the cord. Understanding this helps us distribute mass correctly along the cord's length, which is crucial for calculating gravitational potential energy.

In the problem, we use linear mass density \(\lambda\), defined as \(\lambda = \frac{m}{L}\), where \(m\) is the total mass, and \(L\) is the length of the cord. This allows us to find the mass of each tiny segment of the cord.

• For the cord, \(\lambda = 0.06 \text{ kg/m}\), meaning each meter of the cord weighs 0.06 kg.
• We calculate segment mass by \(dm = \lambda \, dy\), meaning even small slices of the cord at distance \(dy\) from the starting point have a precise mass, allowing accurate calculations of energy.

Linear mass density is vital for correctly applying calculus in mechanical problems like this, ensuring each part's mass is appropriately accounted for in the calculations.