A 20.0 -kg child is on a swing attached to ropes that are \(L=1.50 \mathrm{~m}\) long. Take the zero of the gravitational potential energy to be at the position of the child when the ropes are horizontal. a) Determine the child's gravitational potential energy when the child is at the lowest point of the circular trajectory. b) Determine the child's gravitational potential energy when the ropes make an angle of \(45.0^{\circ}\) relative to the vertical. c) Based on these results, which position has the higher potential energy?

The formula for gravitational potential energy is: \(U = mgh\) where \(U\) = gravitational potential energy \(m\) = mass of the object (in this case, the child) \(g\) = acceleration due to gravity (approximately \(9.81 m/s^2\)) \(h\) = height of the object above the reference point (we'll determine this for each position)

When the ropes are horizontal, the child is at the lowest point of the trajectory. The height \(h_1\) above the reference point (when the ropes are horizontal) is given by the length of the ropes: \(h_1 = L\)

Now we can use the formula for gravitational potential energy at this position: \(U_1 = m \cdot g \cdot h_1\) Plug in the given values: \(U_1 = \left(20.0\,\text{kg}\right) \cdot \left(9.81\, \mathrm{m/s^2}\right) \cdot \left(1.50\, \mathrm{m}\right)\) and find \(U_1\).

We need to determine the child's height \(h_2\) when the ropes make a \(45.0^{\circ}\) angle relative to the vertical. In this position, the child forms a right-angled triangle with the vertical and horizontal sides. We can use the sine function to find the vertical side: \(\sin{45^{\circ}} = \frac{\text{opposite side}}{\text{hypotenuse}} \) Using the given length of the ropes, \(L\), as the hypotenuse: \(\sin{45^{\circ}} = \frac{\text{opposite side}}{1.50\,\text{m}}\) The vertical side will be longer than the horizontal side since it forms a \(45^\circ\) angle with the vertical and is equal to: \(\text{opposite side} = L \cdot \sin{45^{\circ}}\) Therefore, the child's height above the reference point is: \(h_2 = L - \text{opposite side}\)

Now we can use the formula for gravitational potential energy at this position: \(U_2 = m \cdot g \cdot h_2\) Plug in the given values and the height found in Step 4: \(U_2 = \left(20.0\,\text{kg}\right) \cdot \left(9.81\, \mathrm{m/s^2}\right) \cdot h_2\) and find \(U_2\).

Now that we have both gravitational potential energies, \(U_1\) and \(U_2\), we can determine which position has the higher potential energy by comparing their values. The higher value means higher potential energy.