Problem 51 - College Physics

Question

\(\bullet\) A 2.50 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 to air resistance and friction, converting it into thermal energy.

Step 1: Calculate Initial Potential Energy

The initial potential energy when the rock is at an angle of \(11^\circ\) can be calculated using \(PE = mgh\), where \(h = L - L \cos(\theta)\). First, find \(h\):\[h = 1.45\, \mathrm{m} - 1.45\, \mathrm{m} \cos(11^\circ)\]. Then compute the potential energy: \[PE_{\text{initial}} = 2.5\, \mathrm{kg} \times 9.8\, \mathrm{m/s^2} \times h\].

Step 2: Calculate Final Potential Energy

The final potential energy when the rock is at an angle of \(4.5^\circ\) is calculated similarly: \[h' = 1.45\, \mathrm{m} - 1.45\, \mathrm{m} \cos(4.5^\circ)\]. Then compute the potential energy: \[PE_{\text{final}} = 2.5\, \mathrm{kg} \times 9.8\, \mathrm{m/s^2} \times h'\].

Step 3: Calculate Energy Lost

Calculate the energy lost by subtracting the final potential energy from the initial potential energy: \[\text{Energy Lost} = PE_{\text{initial}} - PE_{\text{final}}\].

Step 4: Explain Energy Loss

The lost energy is mainly due to air resistance and friction at the pivot point. These non-conservative forces convert some of the mechanical energy into thermal energy, which dissipates into the surroundings.

Key Concepts

Potential Energy

Potential energy is a type of energy that is stored within an object due to its position relative to a reference point, often the ground. Imagine a rock attached to the end of a rope, like in the exercise we discussed. When the rock is pulled back to an angle and released, it has potential energy because of its elevated position. This energy depends on a few factors:

• Mass of the rock (m): Heavier objects have more potential energy.
• Gravity (g): On Earth, this is approximately 9.8 m/s².
• Height (h): The vertical distance the rock is initially elevated, calculated from the angle and rope length.

The formula to calculate potential energy is given by: \[PE = m imes g imes h\]In our problem, the rock is released from 11 degrees, so we first need to find the height it reaches from this angle. The formula used is: \[h = L - L \cos(\theta)\]where \(L\) is the length of the rope and \(\theta\) is the angle. By plugging these values into our potential energy formula, we can determine how much potential energy the rock initially has.

Energy Loss

Energy loss occurs when the energy in a system decreases as it is transformed from one form to another. In our exercise, the rock on the rope swings back and forth like a pendulum. Initially, it has a certain potential energy. Over time, this energy seems to "disappear" as the rock does not rise to the same angle it started from. There are many different causes for energy loss:

• Friction: When the rope rubs against the pivot point, it converts kinetic energy (movement energy) into thermal energy (heat), which dissipates into the surrounding air.
• Air Resistance: As the rock swings, it cuts through the air, causing drag. This force works against the rock's motion, gradually slowing it down and converting mechanical energy into heat.

All these factors contribute to the energy "lost" from the swing system. By calculating the potential energy at different points (initial and final), we can understand how much energy the system has lost.

Non-conservative Forces

Non-conservative forces are forces where the work done is not recoverable as elastic potential energy. Unlike conservative forces, such as gravity and spring force, which are path-independent and store energy internally, non-conservative forces depend on the path taken and often transform kinetic into thermal energy, among others. In our pendulum exercise, the two primary non-conservative forces at play are friction and air resistance.

• Friction at the pivot: As the rope swings back and forth, it may rub against its pivot point or even stretch slightly. This interaction converts mechanical energy into heat.
• Air resistance: As the rock moves through the air, it faces resistance. This force not only slows down the swing but also dissipates mechanical energy as heat.

These non-conservative forces explain why mechanical energy in many systems, like our swinging rock, is not entirely conserved during motion. The energy is not lost in the universe, but rather transformed into forms that are no longer useful for doing work in this system.