Problem 48 - College Physics

Question

A girl swings back and forth on a swing with ropes that are \(4.00 \mathrm{~m}\) long. The maximum height she reaches is \(2.00 \mathrm{~m}\) above the ground. At the lowest point of the swing, she is \(0.500 \mathrm{~m}\) above the ground. (a) The girl attains the maximum speed (1) at the top, (2) in the middle, (3) at the bottom of the swing. Why? (b) What is the girl's maximum speed?

Step-by-Step Solution

Answer

(a) At the bottom (b) Maximum speed is approximately 5.42 m/s.

Step 1: Understand the Swing Motion

The girl is swinging like a pendulum, moving from a maximum height of 2.00 m to a minimum point 0.500 m above the ground. Her speed varies throughout the swing, being zero at the highest point and reaching maximum at the lowest point.

Step 2: Identify Maximum Speed Point

According to the conservation of energy, the potential energy is highest and kinetic energy is lowest at the top points of the swing, and vice versa at the bottom. Therefore, the girl's speed is maximum at the lowest point because potential energy is converted to kinetic energy.

Step 3: Calculate Gravitational Potential Energy

At the top, the potential energy (PE) is given by \( PE = mgh \), where \( m \) is the mass, \( g = 9.81 \, \text{m/s}^2 \), and \( h = 2.00 - 0.50 = 1.50 \, \text{m} \). At the lowest point, \( PE = 0 \).

Step 4: Apply Energy Conservation

The total mechanical energy is conserved. Thus, the potential energy at the top turns into kinetic energy (KE) at the bottom: \( KE = PE_{\text{top}} = mgh \). Kinetic energy is given by \( KE = \frac{1}{2} mv^2 \).

Step 5: Solve for Maximum Speed

Equate the kinetic energy at the bottom to the potential energy at the top: \( \frac{1}{2} mv^2 = mgh \). Solving for speed \( v \), we get: \[ v = \sqrt{2gh} = \sqrt{2 \times 9.81 \, \text{m/s}^2 \times 1.50 \, \text{m}} \].

Step 6: Compute Numerical Value

Plugging in the values, \( v = \sqrt{2 \times 9.81 \times 1.50} \approx 5.42 \, \text{m/s} \).

Key Concepts

Pendulum Motion

A pendulum is a classic example of a simple harmonic motion, where an object swings back and forth in a regular path. In the case of the girl on the swing, she behaves like a pendulum as she moves from the highest point to the lowest point and back again. The swing moves along an arc-shaped path, with characteristics similar to those of a pendulum in physics experiments.
Several forces act upon the swing, mainly the gravitational force and tension in the ropes. The gravitational force pulls the swing downward, while tension keeps it attached to the pivot point at the top. As the girl swings, her height changes, affecting her potential and kinetic energy. This interplay defines the speed and height dynamics throughout the swing cycle.
Understanding pendulum motion is essential for grasping how energy transforms and transfers during the swing, which relates to the next concepts of gravitational potential and kinetic energy. These principles are foundational in both small-scale experiments and larger systems like amusement park rides.

Gravitational Potential Energy

Gravitational potential energy is a type of energy stored due to an object's position relative to a gravitational source, such as the Earth. In the scenario of the swinging girl, gravitational potential energy is at its peak when she is at the highest point in her swing. This occurs because she is elevated above the ground, influenced by the height variable in the equation:
\[ PE = mgh \]
where:

• \( m \) is the mass of the girl,
• \( g \) is the acceleration due to gravity (approximately 9.81 m/s²),
• \( h \) is the height above the ground.

At the top of her swing, the girl has much potential energy (PE) and zero kinetic energy since she momentarily pauses before swinging back. All her potential energy will convert to kinetic energy as she moves downward, illustrating energy conservation principles.
This conversion is pivotal because it enables the girl to accelerate and eventually achieve her maximum speed at the lowest point of her swing.

Kinetic Energy

Kinetic energy is the energy of motion. For our swinging girl, as she descends from the highest point of the swing, her potential energy begins transforming into kinetic energy. This transformation gains speed, illustrating how energy conversion works dynamically in pendulum motion. When the girl reaches the lowest point, all gravitational potential energy has been converted into kinetic energy.
The kinetic energy (KE) at the lowest point can be calculated using the equation:
\[ KE = \frac{1}{2} m v^2 \]
where:

• \( m \) is her mass,
• \( v \) is her velocity at the lowest point.

The law of conservation of energy tells us that the total energy (sum of potential and kinetic energy) remains constant throughout her swing. By knowing the gravitational potential energy at the higher point, you can solve for the velocity—and thus her kinetic energy—at the lowest point of her swing.
This demonstrates an essential physical principle: energy can change forms—like potential to kinetic—but the total amount remains the same, maintaining the balance of the system.