91Ó°ÊÓ

When we talk about potential energy, we refer to the stored energy an object possesses due to its position or state. In the child's swing example, potential energy is crucial. When the child is lifted to a height with the swing making a 30-degree angle with the vertical, the child gains potential energy due to this elevated position. The formula to calculate potential energy is given by \[ PE = mgh \] where:

• \( m \) is the mass of the child.
• \( g \) is the acceleration due to gravity, usually approximated to \(9.8\, \mathrm{m/s^2}\).
• \( h \) is the height from which the child falls, in this case calculated as \(0.267\, \mathrm{m}\).

This potential energy transforms into kinetic energy as the child swings downward. Understanding potential energy allows us to grasp how energy is conserved and transformed during the swing's motion.

Kinetic energy is the energy possessed by an object due to its motion. When the child on the swing reaches the lowest point in her swing, her potential energy has been mostly converted into kinetic energy. To find the kinetic energy at this point, we can use the formula:\[ KE = \frac{1}{2}mv^2 \]where:

• \( m \) is the mass of the child.
• \( v \) is the velocity of the child at the lowest point of the swing.

In a frictionless scenario, the kinetic energy at the lowest swing point equals the potential energy at the starting height. This transformation showcases the conservation of energy principle, where the total energy remains constant. Given the actual speed of \(2.00\, \mathrm{m/s}\), compared to the theoretical \(2.314\, \mathrm{m/s}\), it's evident that energy is lost, primarily due to friction.

Energy loss due to friction is a common phenomenon where some mechanical energy is transformed into thermal energy because of the resistive forces present. In our swing example, when the child swings down and reaches the lowest point, the speed is not as high as calculated without friction. The calculated speed without friction is \(2.314\, \mathrm{m/s}\), yet the actual speed is only \(2.00\, \mathrm{m/s}\).To find the energy lost due to friction, we calculate the difference in kinetic energy between the frictionless scenario and the actual speed scenario using their respective velocities:\[ \Delta KE = \frac{1}{2}m(v_{calculated}^2 - v_{actual}^2) \]By doing such calculations, we find an energy loss of approximately \(3.81\, \mathrm{J}\).This loss is indicative of the inevitable imperfections in real-world systems, highlighting how various forces like air resistance and friction influence energy conservation.