Question

A mother pulls her daughter, whose mass is \(20.0 \mathrm{~kg}\) and who is sitting on a swing with ropes of length \(3.50 \mathrm{~m}\), backward until the ropes make an angle of \(35.0^{\circ}\) with respect to the vertical. She then releases her daughter from rest. What is the speed of the daughter when the ropes make an angle of \(15.0^{\circ}\) with respect to the vertical?

Short answer

Answer: To find the speed of the child when the ropes make an angle of 15.0° with respect to the vertical, we can apply the conservation of mechanical energy. After calculating the initial and final heights using the given angles, and substituting the known values for mass, gravitational acceleration, and height differences, we can find the final speed of the child. To calculate the speed, use the following formula: $$ v = \sqrt{2(9.81\mathrm{m/s^2})(h_i - h_f)} $$ Plug in the calculated initial and final heights and solve for the final speed, v, to get the answer to the problem.

Step-by-step solution

Identify known and unknown variables

We know the following variables: - Mass of the child, \(m = 20.0\mathrm{~kg}\) - Length of the swing ropes, \(L = 3.50\mathrm{~m}\) - Initial angle with respect to the vertical, \(\theta_1 = 35.0^{\circ}\) - Final angle required, \(\theta_2 = 15.0^{\circ}\) We need to find the final speed, \(v\).

Calculate the initial and final heights

We will first find the initial and final heights of the child with respect to the lowest point of the swing. To do this, we use the following relation for the heights, considering the length of the ropes and the respective angles: $$ h_i = L - L\cos{\theta_1} $$ $$ h_f = L - L\cos{\theta_2} $$ where \(h_i\) is the initial height and \(h_f\) is the final height.

Apply conservation of mechanical energy

Since only gravitational force is acting on the system, mechanical energy is conserved. Therefore, we can write the following equation using the potential energy and kinetic energy of the child at the initial and final instants (\(U\) for potential energy, \(K\) for kinetic energy): $$ K_i + U_i = K_f + U_f $$ Initially, the child is at rest, so the initial kinetic energy is zero. Thus, the equation becomes: $$ U_i = K_f + U_f $$ Potential energy is calculated using the mass, gravitational constant (\(g = 9.81\mathrm{m/s^2}\)) and heights: $$ mgh_i = \frac{1}{2}mv^2 + mgh_f $$ Now, we can solve the equation for the final speed, \(v\).

Calculate the final speed

We can rearrange the equation from step 3 to solve for \(v\): $$ v = \sqrt{2g(h_i - h_f)} $$ Using the values calculated in step 2 for the initial and final heights, and substituting the known values for mass and gravitational acceleration, we can find the final speed, \(v\), when the ropes make an angle of \(15.0^{\circ}\) with respect to the vertical.

Find and report the final value

Plug in the known values and calculate the final speed of the daughter: $$ v = \sqrt{2(9.81\mathrm{m/s^2})(h_i - h_f)} $$ Once the final speed is calculated, it can be reported as the answer to the problem.

Key concepts

Conservation of Energy

In physics, the principle of the conservation of energy tells us that energy cannot be created or destroyed; it can only change from one form to another. This principle is crucial when analyzing the motion of a swing. Here, as the child swings, energy transitions between two main types: potential energy (related to the height of the swing) and kinetic energy (related to its speed).

When the child is initially pulled back, all the energy is stored as potential energy due to her height above the ground. Once released, this energy begins converting into kinetic energy as she descends and speeds up. It's important to remember:

• The total mechanical energy (sum of potential and kinetic energy) remains constant, assuming no external forces like air resistance.
• This allows us to predict speed at different points in her swing by comparing energy states.

Kinetic and Potential Energy

Understanding kinetic and potential energy is crucial in swinging motion. **Potential energy** depends on the height of an object. It's calculated as:\[ U = mgh \]where \(m\) is mass, \(g\) is gravity (\(9.81 \mathrm{~m/s^2}\)), and \(h\) is height from a reference point. When the child is at the peak of her swing, potential energy is maximized.

When the swing is in motion, potential energy transforms into **kinetic energy**, described by:\[ K = \frac{1}{2}mv^2 \]Here, \(v\) is the speed. As the child descends, potential energy decreases and converts to kinetic energy, increasing her speed.

Remember:

• Kinetic energy is highest at the lowest point of the swing when speed is maximum.
• Potential energy is highest at the peak when the swing briefly stops before reversing direction.

Swing Motion Analysis

Analyzing the motion of a swing involves understanding how energy changes throughout the swing's path. Initially, when the mother pulls the swing back, the child has maximum potential energy. As the swing is released and the child moves downward, this energy starts converting into kinetic energy.

The angles at different stages of the swing help determine the swing’s height and subsequently, its speed. By knowing the difference in height between two positions (using angles), you can calculate speed using conservation of energy principles.

• At an angle of \(15^{\circ}\) to the vertical, the child’s speed can be determined from the change in height from the initial position.
• The formula \( v = \sqrt{2g(h_i - h_f)} \) is used, ensuring easy calculation once the heights corresponding to the initial (\(35^{\circ}\)) and final angles are obtained.

This height difference translates directly into the speed of the swing at different angles, illustrating the energy transformation in motion.