Problem 79 - College Physics

Question

\(\bullet\) \(\bullet\) A wooden rod of negligible mass and length 80.0 \(\mathrm{cm}\) is pivoted about a horizontal axis through its center. A white rat with mass 0.500 kg clings to one end of the stick, and a mouse with mass 0.200 kg clings to the other end. The system is released from rest with the rod horizontal. If the animals can manage to hold on, what are their speeds as the rod swings through a vertical position?

Step-by-Step Solution

Answer

The speed is approximately 2.8 m/s for both the rat and the mouse.

Step 1: Identify Key Information

First, we need to identify the key parameters of the problem: the length of the rod is 80.0 cm, which is equivalent to 0.8 meters. The mass of the white rat is 0.500 kg, and the mass of the mouse is 0.200 kg. The rod is pivoted about its center, meaning each animal is 0.4 meters from the pivot point.

Step 2: Understand Energy Conservation

Since the system is released from rest with the rod horizontal, we will use the Conservation of Energy principle to solve the problem. Initially, when the rod is horizontal, the potential energy is at its maximum, and the kinetic energy is zero. As the rod swings to a vertical position, the potential energy is converted into kinetic energy.

Step 3: Calculate Initial Potential Energy

The initial potential energy (PE_initial) is the sum of the gravitational potential energies of the two animals relative to the lowest point they will reach. This is calculated as:\[ PE_{initial} = m_{rat}gh_{rat} + m_{mouse}gh_{mouse} \] where \( h_{rat} = h_{mouse} = \frac{0.8}{2} \) meters because each animal is lifted by the height of half of the rod's length. Substitute and calculate:\[ PE_{initial} = (0.500 \times 9.8 \times 0.4) + (0.200 \times 9.8 \times 0.4) \]

Step 4: Calculate Kinetic Energy at Vertical Position

The kinetic energy (KE) of the system when the rod reaches a vertical position is equal to the initial potential energy because energy is conserved:\[ KE = \frac{1}{2} m_{rat} v_{rat}^2 + \frac{1}{2} m_{mouse} v_{mouse}^2 \] Since both animals are at the same height, they will have the same speed \( v \) at the vertical position. Thus:\[ 0.5(0.500 + 0.200)v^2 = PE_{initial} \]

Step 5: Solve for the Speed

Plug the value of \( PE_{initial} \) we get from Step 3 into the equation from Step 4 and solve for \( v \):\[ v = \sqrt{\frac{2 \times PE_{initial}}{0.500 + 0.200}} \]Calculate this value to find \( v \).

Key Concepts

Conservation of Energy

The principle of conservation of energy is a fundamental idea in physics that states energy cannot be created or destroyed, only transformed from one form to another. This concept is crucial when analyzing systems like our exercise involving the rod and animals. Here, the system initially possesses gravitational potential energy because it starts from rest with the rod horizontal.

As the rod swings down, the potential energy is converted into kinetic energy. By the time the rod reaches a vertical position, nearly all of the potential energy has been transformed. This makes energy conservation a powerful tool in solving physics problems since it allows us to equate the initial potential energy to the kinetic energy at any other point without needing to depend on forces or accelerations.

This means we start by calculating all the potential energy held by the animals when the rod is horizontal. As they swing to the vertical position, this energy is converted into the energy of motion, also known as kinetic energy.

Gravitational Potential Energy

Gravitational potential energy (GPE) refers to the energy held by an object because of its position in a gravitational field. In our scenario, this applies to the white rat and the mouse at the ends of the rod.

GPE is given by the formula:

• \( PE = mgh \)

Where:

• \( m \) = mass of the object
• \( g \) = acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \))
• \( h \) = height above the lowest point

In the exercise, each animal starts at a height equal to half the length of the rod due to its pivot at the center. As the rod falls, this height decreases to zero as the potential energy converts into kinetic energy.

By considering both masses and their heights, we can compute the system's initial gravitational potential energy, which will be useful in solving for the speed in later calculations.

Kinetic Energy

Kinetic energy is the energy associated with the motion of an object. When the rod with the animals attached swings to a vertical position, all initial gravitational potential energy has been transformed into kinetic energy.

This type of energy is calculated using:

• \( KE = \frac{1}{2}mv^2 \)

Where:

• \( m \) = mass of the moving object
• \( v \) = speed of the object

In our case, both the white rat and the mouse will have the same speed since they are both revolving around the pivot at the same radial distance. Their combined kinetic energy is equal to the potential energy the system initially had.

By using conservation principles, we can set the potential energy equal to the kinetic energy and solve for the velocity of these animals as they reach the lowest point of their swing.

Physics Problem Solving

Physics problem solving often requires a methodical approach, using known principles to find the unknowns. In our exercise, the key principle to use is the conservation of energy. Here’s a simple guide to approach such problems:

• **Identify Given Information:** Begin by extracting all given information from the problem. Like rod length, masses, and pivot points.
• **Choose the Right Principles:** Decide which physics principles are applicable. In this case, conservation of energy is ideal because no external forces act beyond gravity.
• **Translate Words to Equations:** Turn the problem’s description into mathematical equations. Write down the equations for initial and final energy states.
• **Solve Mathematically:** Substitute values into the equations and solve for the unknown. Step-by-step calculations will help in tracking the solution efficiently.
• **Check Plausibility:** Always check if the obtained results make sense physically.

Applying such a systematic approach ensures profound understanding and correct solutions. It also enhances the student's problem-solving skills, critical for tackling various physics questions effectively.