Question

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?

Short answer

Answer: To determine which position has a higher gravitational potential energy, first calculate the child's height at each position by following Steps 2 and 4. Then, calculate the gravitational potential energy at each position using Steps 3 and 5. Finally, compare the potential energies and determine which position has the higher potential energy by following Step 6.

Step-by-step solution

Understand gravitational potential energy formula

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)

Calculate the child's height when the ropes are horizontal

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: \(h_1 = L\)

Calculate gravitational potential energy when ropes are horizontal

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 \(U_1\).

Find the child's height when ropes make a \(45.0^{\circ}\) angle

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: \(h_2 = L - \text{opposite side}\)

Calculate gravitational potential energy when ropes make a \(45.0^{\circ}\) angle

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 \(U_2\).

Compare potential energies and determine which position has higher potential energy

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.

Key concepts

Pendulum Motion

Pendulum motion describes the back-and-forth swinging of an object attached to a pivot point, like a swing or a simple pendulum. This motion is significant because it exemplifies periodic motion, which means it repeats itself at regular intervals. In the context of a child swinging, the pivot is the point where the ropes are attached to a swing set. When the child is at the highest point of the swing, they momentarily stop before changing direction, indicating the interplay between potential and kinetic energies. At the lowest point, the speed is the highest, and all the potential energy is converted to kinetic energy, showing energy conservation in action. The motion of a pendulum in such a scenario follows a path known as a circular trajectory with a radius equivalent to the length of the swing's ropes. Understanding pendulum motion allows us to explore concepts like gravitational potential energy and kinetic energy, which are fundamental in describing how objects move within a gravitational field.

Energy Conservation

Energy conservation is a crucial principle in physics, stating that the total energy in a closed system remains constant over time. When discussing a swinging pendulum, this principle is illustrated by the transformation of energy between gravitational potential energy and kinetic energy. - **Gravitational Potential Energy (GPE):** At the peak of its swing, the pendulum holds maximum potential energy due to its elevated position. - **Kinetic Energy (KE):** At the lowest point of the swing, the pendulum's speed—and consequently, its kinetic energy—is at its maximum because the gravitational potential energy has been completely transformed into kinetic energy. During the pendulum's motion, energy moves back and forth between these two forms. Potential energy is highest at the highest point, and kinetic energy is highest at the lowest point.

Trigonometry in Physics

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is especially important in physics for solving problems involving angles, such as those found in pendulum motion.In the given exercise, understanding the child's position at various swing points involves using trigonometric functions like the sine function. When the child swings and makes an angle with the vertical, it creates a right triangle. The length of the pendulum rope serves as the hypotenuse.- **Finding Height Using Sine:** To find the vertical distance (height) from the reference point when the ropes are at an angle, you can use \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \).- By substituting the known values, it is possible to solve for the opposite side, which then helps compute the gravitational potential energy at different positions in the swing.By incorporating trigonometry, one can accurately compute the height differences in pendulum problems, crucial for calculating potential energy changes.