Problem 59 - College Physics

Question

A block with a mass \(m_{1}=6.0 \mathrm{~kg}\) sitting on a frictionless table is connected to a suspended mass \(m_{2}=2.0 \mathrm{~kg}\) by a light string passing over a frictionless pulley. Using energy considerations, find the speed at which \(m_{2}\) hits the floor after descending \(0.75 \mathrm{~m}\). (Note: A similar problem in Example 4.6 was solved using Newton's laws.)

Step-by-Step Solution

Answer

The speed of \(m_2\) when it hits the floor is approximately 3.42 m/s.

Step 1: Identify Energy Conservation Principle

In the given system, we apply the principle of conservation of mechanical energy. This principle states that the total mechanical energy (sum of potential and kinetic energy) remains constant if there are no non-conservative forces (like friction) doing work.

Step 2: Determine Initial Energy

Initially, when the mass \(m_2\) is about to fall, it has gravitational potential energy and both masses have no kinetic energy because they start from rest. The initial potential energy \(PE_{initial}\) of \(m_2\) is \(m_2 g h\), where \(h = 0.75\) m.

Step 3: Set Final Energy Components

When \(m_2\) has fallen 0.75 m, it has no potential energy (as it's at the lowest point) but both masses have kinetic energy. The final kinetic energy \(KE_{final}\) is \(\frac{1}{2} m_1 v^2 + \frac{1}{2} m_2 v^2\), where \(v\) is the velocity we need to find.

Step 4: Apply Energy Conservation Equation

Set the initial energy equal to the final energy. Thus, \(m_2 g h = \frac{1}{2} m_1 v^2 + \frac{1}{2} m_2 v^2\). Simplify the expression to \(m_2 g h = \frac{1}{2} (m_1 + m_2) v^2\).

Step 5: Solve for \(v\)

Plug in the values: \(m_1 = 6\) kg, \(m_2 = 2\) kg, \(g = 9.8\) m/s², and \(h = 0.75\) m into the equation \(m_2 g h = \frac{1}{2} (m_1 + m_2) v^2\). Solve for \(v\): \[v = \sqrt{\frac{2 m_2 g h}{m_1 + m_2}} = \sqrt{\frac{2 \times 2 \times 9.8 \times 0.75}{6 + 2}}\approx 3.42 \text{ m/s}\]

Key Concepts

Mechanical Energy

Mechanical energy is the combination of potential and kinetic energy in a system. In any physical system, if there are no external forces like friction or air resistance, the total mechanical energy remains constant. This is what's happening in the described exercise, where a block and a suspended mass interact through a pulley. At the start, the system's energy is entirely potential, as the block is yet to move. As the suspended mass falls and gains speed, some of the potential energy converts into kinetic energy. The sum of both energies stays the same the whole time, thanks to the conservation of mechanical energy principle. This principle simplifies solving many physics problems as it allows us to track the energy transfers without delving deeply into forces and accelerations directly. Understanding mechanical energy helps bridge the gap between abstract concepts and real-world physics scenarios.

• Mechanical energy = potential energy + kinetic energy
• Total mechanical energy stays the same if no external forces are acting

Potential Energy

Potential energy is the stored energy in an object due to its position or state. In our exercise, the potential energy is specifically gravitational, because it depends on the height of the suspended mass above the ground. This energy can be calculated with the formula: \[PE = mgh\] where:

• \(m\) is the mass
• \(g\) is the acceleration due to gravity \( (9.8 \, \text{m/s}^2) \)
• \(h\) is the height

When the mass initially hangs at \(0.75 \text{ m}\), it has gravitational potential energy because it has the capacity to do work on the way down. As the mass descends, this potential energy is converted to kinetic energy, allowing the mass to accelerate as it falls. The beauty of potential energy lies in its ability to be converted into other energy forms, driving motion and change.

Kinetic Energy

Kinetic energy is the energy of motion. Any object in motion possesses kinetic energy, which can be computed using the equation: \[ KE = \frac{1}{2}mv^2 \] where:

• \(m\) is the mass
• \(v\) is the velocity

In a frictionless setup like our exercise, when the block on the table and the descending mass start moving, they gain kinetic energy. Initially, the system has no kinetic energy, as the two masses are still. As the suspended mass falls, it speeds up, causing both masses to acquire kinetic energy. The kinetic energy gained balances the potential energy lost, keeping the total mechanical energy constant. This conversion demonstrates how energy can transform from one type to another while keeping the overall energy level unchanged in a closed system.

Frictionless Systems

In physics, a frictionless system is an idealized model where no energy is lost to friction. This means all energy changes are internal and track smoothly between different forms, like potential and kinetic energy. Our problem uses a frictionless table and pulley system to focus solely on energy conservation principles without other complicating factors. In the real world, friction is common and saps energy from many processes, turning mechanical energy into heat or sound. However, removing friction from a problem simplifies equations and lets us apply energy conservation principles without worrying about energy losses. This makes frictionless systems powerful tools for learning and understanding basic dynamics and energy transformations in physics.