Problem 95 - University Physics with Modern Physics

Question

Two objects with masses 5.00 \(\mathrm{kg}\) and 2.00 \(\mathrm{kg}\) hang 0.600 \(\mathrm{m}\) above the floor from the ends of a cord 6.00 \(\mathrm{m}\) long passing over a frictionless pulley. Both objects start from rest. Find the maximum height reached by the 2.00 -kg object.

Step-by-Step Solution

Answer

The 2.00 kg object reaches a maximum height of 1.20 m.

Step 1: Understand the System

We have two objects connected by a cord over a pulley. The first object has a mass of 5.00 kg and the second object has a mass of 2.00 kg. The pulley is frictionless, meaning no energy is lost. Initially, both objects start from rest.

Step 2: Identify the Forces

The 5.00 kg object will exert a greater gravitational force than the 2.00 kg object, causing the 5.00 kg object to move downward and the 2.00 kg object to move upward.

Step 3: Set up the Energy Equation

Use the principle of conservation of energy. Initially, the only energy in the system is potential energy from both masses. As the system moves, gravitational potential energy changes to kinetic energy.

Step 4: Potential Energy Change

The change in potential energy when the 5.00 kg object moves down (and the 2.00 kg object moves up) is given by:\[ \Delta U = m_1 g h_1 - m_2 g h_2 \]where \(m_1 = 5.00 \, \mathrm{kg}\), \(m_2 = 2.00 \, \mathrm{kg}\), and \(h_1 = h_2 = x\) since both move the same distance.

Step 5: Kinetic Energy Conversion

All the potential energy lost by the 5.00 kg object converts to kinetic energy in the system:\[ \Delta U = \Delta K = \frac{1}{2} (m_1 + m_2) v^2 \]Solve for \(v^2\) to find the velocity at equilibrium before stopping due to the 2.00 kg block reaching max height.

Step 6: Calculate Maximum Height

After solving, use the velocity to find the new height of the 2.00 kg object using the energy conversion equations to find potential energy at this peak height:\[ m_2 g h_{max} = \frac{1}{2} m_1 g h + \frac{1}{2} m_2 g h \]

Step 7: Solve the Equations

Substitute and solve for maximum height \(h_{max}\). You will find the 2.00 kg object reaches a height of 1.20 m.

Key Concepts

Pulley Systems

Pulley systems are fundamental in mechanics, often used to lift or move loads with reduced effort. In a simple pulley system like the one described in our exercise, two masses are connected by a cord over a pulley. The pulley's role is to change the direction of the force applied, allowing objects on either side to move in opposite directions. This pulley is frictionless, meaning it does not lose energy due to friction, making calculations simpler and focused on the transformation of energy. Pulley systems can be classified based on the number of wheels or pulleys used, like simple pulleys with one wheel or compound ones with multiple. The effectiveness of a pulley system is often measured by its mechanical advantage, which helps in determining how much easier a task is using the system compared to direct lifting. In our system, as one object goes up, the other comes down - this seesaw-like motion is integral to understanding how potential and kinetic energy interchange.

Gravitational Potential Energy

Gravitational potential energy is a type of energy an object possesses because of its position relative to the Earth's surface. When an object is at a height, it possesses energy that has the potential to do work because of the force of gravity acting on it.The formula for gravitational potential energy is given by:\[ U = mgh \]where:

• \( m \) is the mass of the object,
• \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \) at the Earth's surface),
• \( h \) is the height above the reference point.

In our exercise, the potential energy of each mass changes as the 5.00 kg object descends and the 2.00 kg object ascends. This change in height results in a transfer of energy, showcasing the conservation principle where potential energy decreases in one object and the increase in another manifests as kinetic energy in the system overall.

Kinetic Energy

Kinetic energy is the energy an object possesses because of its motion. Whenever an object moves, it has kinetic energy, which depends on both its mass and velocity. The standard formula for kinetic energy is:\[ K = \frac{1}{2}mv^2 \]where:

• \( m \) is the mass of the moving object,
• \( v \) is the velocity of the object.

As the 5.00 kg object starts descending, its potential energy converts into kinetic energy, increasing the system's overall kinetic energy. The speed gained by the object is essentially the result of this energy conversion. In pulley systems, kinetic energy is shared between the connected objects. When the object with a greater mass descends, its lost potential energy is matched by a gain in kinetic energy, which in turn, provides energy to the smaller mass, allowing it to reach maximum height after the velocity energy has been fully optimized. This showcases how energy shifts within a system, and why understanding kinetic energy is crucial for predicting motions.