Problem 2 - University Physics with Modern Physics

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 J. At the highest position of its 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 \(=8 \mathrm{~J}\) e) kinetic energy \(=8 \mathrm{~J}\) and gravitational potential energy \(=4\) J.

Step-by-Step Solution

Answer

Answer: At the highest position of the swing, the kinetic energy is 0 J, and the gravitational potential energy is 12 J.

Step 1: Calculate the total mechanical energy at the bottom of the swing

Since energy is conserved, we can calculate the total mechanical energy at the bottom of the swing by adding the given kinetic energy and gravitational potential energy. In this case, KE = 8 J and PE = 4 J. Total Mechanical Energy (TME) = KE + PE = 8 J + 4 J = 12 J

Step 2: Determine energies at the highest position of the swing

At the highest position of the swing, the pendulum is momentarily at rest, which means its kinetic energy is zero (KE = 0 J). When the kinetic energy is 0 J, all the energy must be potential energy, according to the conservation of mechanical energy. So, the gravitational potential energy at the highest position should be equal to the total mechanical energy we calculated in step 1. So, the energies at the highest position of the swing are: KE = 0 J PE = 12 J

Step 3: Choose the correct answer

Now, we can compare our findings with the given options to choose the correct answer. Our KE and PE values match with option (c). The correct answer is (c) kinetic energy \(=0 \mathrm{~J}\) and gravitational potential energy \(=12 \mathrm{~J}\).

Key Concepts

Mechanical Energy

Mechanical energy is the total energy of an object derived from its motion and position. It consists of two main types of energy: kinetic and potential. When studying systems like a swinging pendulum, it is important to consider how these energies interact and change.

Mechanical energy is conserved in a closed system, meaning it doesn't gain or lose energy unless acted upon by an external force. In the case of the pendulum, the total mechanical energy remains constant as it swings. This means if kinetic energy increases, potential energy decreases and vice versa.

The conservation of mechanical energy helps us predict how high the pendulum will swing and when it will be at its fastest. By knowing the total mechanical energy, we can determine the distribution of kinetic and potential energy at any point in the pendulum’s path. In our exercise, even though the energies change as the pendulum moves, the sum of kinetic and potential energy remains at 12 J throughout the motion.

Potential Energy

Potential energy is the stored energy of an object due to its position or state. In the example of a pendulum, gravitational potential energy is due to its height relative to the lowest point of the swing.

Potential energy can be calculated using the formula:\[PE = mgh\]where:

• \(PE\) is potential energy,
• \(m\) is the mass of the object,
• \(g\) is the acceleration due to gravity,
• \(h\) is the height of the object.

For our pendulum, as it rises, it gains potential energy since its height increases.

At the highest point of the swing, it has maximum potential energy and minimum kinetic energy. This is consistent with our exercise where at the top, the potential energy was calculated to be 12 J. This value reflects the total mechanical energy when the pendulum momentarily stops before reversing its motion.

Kinetic Energy

Kinetic energy refers to the energy an object has due to its motion. It reflects how fast an object is moving and can be calculated by:\[KE = \frac{1}{2} mv^2\]where:

• \(KE\) is kinetic energy,
• \(m\) is the mass of the object,
• \(v\) is the velocity of the object.

In a pendulum, as it swings downwards, it converts potential energy to kinetic energy, speeding up as it loses height.

At the bottom of its swing, the pendulum’s kinetic energy is at its peak since its speed is greatest there. This was confirmed in our problem where kinetic energy was 8 J at the lowest point.

As the pendulum ascends, kinetic energy decreases while potential energy rises until it comes to a brief stop at its highest point with zero kinetic energy.This exchange of energy ensures that the sum of kinetic and potential energy remains constant, as seen in the different phases of the pendulum’s motion.