A 40.0-N child is in a swing that is attached to ropes 2.00...

A $40.0-\mathrm{N}$ child is in a swing that is attached to ropes $2.00 \mathrm{~m}$ long. Find the gravitational potential energy of the child-Earth system relative to the child's lowest position when (a) the ropes are horizontal, (b) the ropes make a $30.0^{\circ}$ angle with the vertical, and (c) the child is at the bottom of the circular arc.

Key Concepts

Reference Level in Potential Energy Calculations

When working with potential energy, it is necessary to choose a reference level where the potential energy is defined to be zero. In many problems, including those involving pendular motion, the lowest point in the motion is chosen as the reference level. This helps simplify calculations by measuring all changes in potential energy relative to that point.

Geometry of Circular Motion in a Pendulum

In scenarios involving pendulums or circular arcs, the vertical displacement from the lowest point is calculated using trigonometric relationships. With a fixed length L and an angular displacement, the vertical height increase is determined by the difference between the full length and the vertical component of the displaced position, typically expressed as h = L - L cos?. This relationship is essential for calculating the change in gravitational potential energy as the object moves along its circular path.

Gravitational Potential Energy

This concept refers to the energy that an object possesses because of its position within a gravitational field. It is defined by the equation U = mgh, where m is the object's mass, g is the acceleration due to gravity, and h is its height above a defined reference level. This energy quantifies how much work would be done by gravity if the object were to fall from that height.

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