Question

A pendulum swings in a vertical plane. At the bottom of the swing, the kinetic energy is \(8 \mathrm{~J}\) and the gravitational potential energy is \(4 \mathrm{~J}\). At the highest position of the swing, the kinetic and gravitational potential energies are a) kinetic energy \(=0 \mathrm{~J}\) and gravitational potential energy \(=4 \mathrm{~J}\). b) kinetic energy \(=12 \mathrm{~J}\) and gravitational potential energy \(=0 \mathrm{~J}\). c) kinetic energy \(=0 \mathrm{~J}\) and gravitational potential energy \(=12 \mathrm{~J}\) d) kinetic energy \(=4\) J and gravitational potential energy \(=8\) J. e) kinetic energy \(=8 \mathrm{~J}\) and gravitational potential energy \(=4 \mathrm{~J}\)

Short answer

Answer: Kinetic energy = 0 J and gravitational potential energy = 12 J.

Step-by-step solution

Write the conservation of mechanical energy equation

The conservation of mechanical energy states that the total mechanical energy of a closed system remains constant if only conservative forces (such as gravity) are acting upon it. The total mechanical energy is the sum of kinetic energy and gravitational potential energy. Therefore, we can write the equation as follows: Total mechanical energy at the bottom = Total mechanical energy at the top

Determine the total mechanical energy at the bottom

According to the given information, at the bottom of the swing, the kinetic energy (KE) is 8 J and the gravitational potential energy (GPE) is 4 J. We can find the total mechanical energy at the bottom: Total mechanical energy at the bottom = KE + GPE = 8 J + 4 J = 12 J

Apply the conservation of mechanical energy equation to find the correct option

We will now evaluate each option and determine if the total mechanical energy at the top is equal to 12 J: a) KE = 0 J and GPE = 4 J: Total mechanical energy = 0 J + 4 J = 4 J (This option is incorrect) b) KE = 12 J and GPE = 0 J: Total mechanical energy = 12 J + 0 J = 12 J (This option is incorrect, since the pendulum should have 0 kinetic energy at the highest point) c) KE = 0 J and GPE = 12 J: Total mechanical energy = 0 J + 12 J = 12 J (This option is correct) d) KE = 4 J and GPE = 8 J: Total mechanical energy = 4 J + 8 J = 12 J (This option is incorrect, since the pendulum should have 0 kinetic energy at the highest point) e) KE = 8 J and GPE = 4 J: Total mechanical energy = 8 J + 4 J = 12 J (This option is incorrect, since the pendulum should have 0 kinetic energy at the highest point) The correct answer is (c) kinetic energy \(=0 \mathrm{~J}\) and gravitational potential energy \(=12 \mathrm{~J}\).

Key concepts

Kinetic Energy

Kinetic energy (KE) is the energy that an object possesses due to its motion. It can be expressed mathematically with the formula:
\[\begin{equation} KE = \frac{1}{2}mv^2 \[5pt]\end{equation}\]where \(m\) is the mass of the object and \(v\) is its velocity. In the context of a swinging pendulum, the kinetic energy is greatest at the lowest point of the swing, where the velocity is highest.

• When the pendulum reaches the very bottom of its arc, all of its energy is kinetic because it is moving the fastest at this point.
• As the pendulum rises, it slows down, so its kinetic energy decreases.

When analyzing problems involving kinetic energy, it's crucial to consider the conservation of energy, which indicates that in a closed system, energy can neither be created nor destroyed. Therefore, the kinetic energy transformation is a key component in such physics problems.

Gravitational Potential Energy

Gravitational potential energy (GPE) refers to the energy an object has because of its position in a gravitational field. The Earth's gravity provides a well-known gravitational field, and the potential energy can be calculated using the formula:
\[\begin{equation} GPE = mgh \[5pt]\end{equation}\]where \(m\) is the mass of the object, \(g\) is the acceleration due to gravity, and \(h\) is the height above the reference point.

• In the case of a pendulum, the potential energy is highest at the peak of its swing.
• The potential energy is converted to kinetic energy as the pendulum swings downward.

At the highest point of the pendulum's arc, the velocity is zero, thus the kinetic energy is zero, and all the energy is gravitational potential energy. This reciprocal transformation between kinetic and potential energy is a hallmark of pendular motion.

Pendulum

A pendulum is a classic example of an oscillating system, where physics students can readily observe the interplay between kinetic energy and gravitational potential energy. The pendulum moves back and forth in a regular motion, called simple harmonic motion, due to the force of gravity acting on it.

• The length of the pendulum is a crucial factor determining its period of oscillation – the time it takes to complete one full swing.
• As it swings, it exchanges kinetic and potential energy, without gaining or losing any total mechanical energy when no external forces like friction are acting upon it.
• At the highest points of its swing (its amplitude), it has maximum potential energy and minimum kinetic energy.
• At the lowest point of the swing, the pendulum has its maximum kinetic energy and minimum potential energy.

Through the study of pendulums, students gain insights into the concepts of energy conservation, forces, and the influence of gravity on motion.

Conservative Forces

Conservative forces, such as gravity, are those that allow for the conservation of mechanical energy within a system. When only conservative forces do work on an object, the total mechanical energy (the sum of kinetic and potential energy) remains constant.

• Gravity is the conservative force acting on a pendulum.
• In the absence of non-conservative forces like friction and air resistance, a pendulum's total mechanical energy would be conserved.
• The work done by conservative forces is path-independent, which means the total work done only depends on the start and end points, and not the path taken.

Using the concept of conservative forces, one can predict the behavior of systems like pendulums, roller coasters, or planets orbiting the Sun, knowing that if energy is not lost to outside forces, it will merely transform rather than diminish.