Problem 49 - College Physics

Question

A \(2.50 \mathrm{~kg}\) rock is attached at the end of a thin, very light rope \(1.45 \mathrm{~m}\) long and is started swinging by releasing it when the rope makes an \(11^{\circ}\) angle with the vertical. You record the observation that it rises only to an angle of \(4.5^{\circ}\) with the vertical after \(10 \frac{1}{2}\) swings. (a) How much energy has this system lost during that time? (b) What happened to the "lost" energy? Explain how it could have been "lost."

Step-by-Step Solution

Answer

The system lost energy primarily due to friction and air resistance.

Step 1: Calculate Initial and Final Heights

To find the energy lost, we first calculate the initial and final heights.- **Initial Height:** When the rock is at an angle of \(11^{\circ}\), the initial vertical height \(h_i\) from the lowest point is given by: \[ h_i = L (1 - \cos(\theta_i)) \] where \(L = 1.45 \ m\) and \(\theta_i = 11^{\circ}\).- **Final Height:** When the rock is at an angle of \(4.5^{\circ}\), the final height \(h_f\) is: \[ h_f = L (1 - \cos(\theta_f)) \] where \(\theta_f = 4.5^{\circ}\).By substituting the values into these formulas, we can find both heights.

Step 2: Compute Initial and Final Potential Energies

With the initial and final heights calculated, we can determine the potential energies using:- **Initial Potential Energy (PE):** \[ PE_i = mgh_i \] where \(m = 2.50\ kg\) (mass of the rock) and \(g = 9.81\ m/s^2\) (acceleration due to gravity).- **Final Potential Energy (PE):** \[ PE_f = mgh_f \]Calculate \(PE_i\) and \(PE_f\).

Step 3: Calculate the Energy Lost

Energy lost is simply the difference between the initial and final potential energies:\[ \Delta E = PE_i - PE_f \]Plug the values of \(PE_i\) and \(PE_f\) obtained from Step 2 into this formula to find \(\Delta E\).

Step 4: Explain Energy Loss

Energy is 'lost' primarily due to non-conservative forces like air resistance and friction at the pivot point that convert mechanical energy into thermal energy and sound. Over time, as the pendulum swings, these forces continuously extract energy from the system.

Key Concepts

Pendulum Motion

A pendulum is a simple system often used in physics to study motion and energy concepts. It consists of a weight or bob attached to the end of a string or rod, fixed at a pivot point. This system oscillates back and forth under the influence of gravity. The nature of its motion can help us understand conservation of energy in a mechanical system.

In a perfect world, with no external forces acting, a pendulum would continue swinging forever, maintaining its energy. The pendulum's motion converts potential energy (when at the highest points) into kinetic energy (when passing through the lowest point). However, in the real world, imperfections like air resistance affect this perfect exchange.

Energy Loss

Energy loss in a pendulum system occurs due to several factors that introduce resistance to its motion. Over time, a swinging pendulum doesn't reach the same height it initially did because of energy being used up to overcome these forces.

Some common reasons for energy loss include:

• Air resistance: The pendulum swings through air, which provides a frictional force that dissipates energy.
• Friction at the pivot: The point where the pendulum is attached can also have friction, further reducing energy.
• Sound: Minor amounts of energy are converted into sound as the pendulum swings.

Energy is never destroyed; it merely changes from one form to another. In the context of a pendulum, mechanical energy is usually transformed into thermal energy and sound, leading to the gradual loss of the amplitude of oscillation.

Potential Energy

Potential energy in a pendulum system is at its maximum when the pendulum is at the highest point in its swing. This energy is determined by the height of the pendulum bob relative to its lowest point and its mass.

The potential energy (PE) can be calculated using the formula: \[ PE = mgh \] where:

• \( m \) is the mass of the pendulum bob
• \( g \) is the acceleration due to gravity
• \( h \) is the height above the lowest point

The beauty of potential energy lies in its ability to be completely transformed into kinetic energy, which occurs as the pendulum swings downward. This transformation between potential and kinetic energy is what keeps the pendulum moving in its arc unless energy is lost to other forces.

Mechanical Energy Dissipation

Mechanical energy dissipation refers to the process in which mechanical energy of a system is transformed into other forms of energy, such as heat or sound energy. In our pendulum scenario, mechanical energy (the sum of kinetic and potential energy) is not conserved due to dissipation.

When a pendulum begins to swing, it starts with a certain amount of mechanical energy. As it progresses through each swing, mechanical energy dissipates because of non-conservative forces like air resistance and pivot friction. This energy transformation reduces the amplitude of each swing, causing the pendulum to eventually come to a stop if no external energy is added back to the system.

Non-Conservative Forces

Non-conservative forces differ from conservative forces in that they change the total mechanical energy of a system. While conservative forces like gravity allow for the reversible transfer of energy between potential and kinetic forms, non-conservative forces result in energy conversion to non-recoverable forms, such as heat.

In a pendulum, non-conservative forces include:

• Frictional force: Both in air and at the pivot point.
• Sound energy: Energy that dissipates as noise.

These forces drain energy away from the system, meaning that less energy is available for the pendulum to swing back to its original height. Consequently, each swing is slightly shorter than the one before, highlighting the importance of accounting for non-conservative forces when examining real-world pendulum motion.