Problem 1 - Applied Mathematics

Question

Two particles \(\mathrm{A}\) and \(\mathrm{B}\) are connected by a light inelastic string which passes over a smooth pulley. A is of mass \(m\) and \(B\) is of mass \(2 m .\) Initially both particles are at rest at a depth \(2 l\) below the pulley. If they are released from rest find their velocity when each has moved a distance \(l\).

Step-by-Step Solution

Answer

The velocity of each particle when they have moved a distance \(l\) is \(\sqrt{2gl}\r).

Step 1: Identify given values

Given: Mass of A is \(m\), Mass of B is \(2m\), Depth below pulley is \(2l\). Initial velocity is 0 and distance moved is \(l\). Find the velocity of each particle when they have moved \(l\).

Step 2: Apply the principle of conservation of energy

As the system is released and moves, the potential energy lost by particle B is converted into kinetic energy in the system. Write the conservation of energy equation: \[ mgh_1 + \frac{1}{2}mv^2_1 = mgh_2 + \frac{1}{2}mv^2_2 \] Note that since both particles are at the same height so initial and final potential energy of both particles can be joined together. It can also be written as: \[ mgh = \frac{1}{2} (2m + m )v^2 \] where \(h\) is the height moved and \(v\) is the velocity both particles move with.

Step 3: Calculate the change in potential energy

Particle A moves up by \(l\) and particle B moves down by \(l\), changing their potential energy. Use the potential energy formula \(U = mgh\).

Step 4: Set up and solve energy conservation equation

Equate the potential energy lost by particle B to the kinetic energies of both particles: \[ mgl + 2mgl = \frac{1}{2}(m + 2m)v^2 \] Simplify the equation: \[ 3mgl = \frac{3mv^2}{2} \]

Step 5: Solve for the final velocity

Rearrange to isolate \(v\): \[ 6mgl = 3mv^2 \] Simplify and solve: \[ 6gl = 3v^2 \] \[ 2gl = v^2 \] \[ v = \sqrt{2gl} \]

Key Concepts

Potential Energy

Potential energy is a type of stored energy that an object possesses due to its position or state. In this exercise, we are dealing with gravitational potential energy. This energy depends on the height of an object above a reference point, usually the ground, and can be calculated using the formula: \( U = mgh \) where \( U \) is the potential energy, \( m \) is the mass, \( g \) is the acceleration due to gravity, and \( h \) is the height.

In our exercise, particle A and particle B change their height as they move, resulting in changes to their potential energy. When both particles move a distance \( l \), the height changes, leading to a change in potential energy for each particle.

Key points to remember about potential energy:

• The higher an object is, the more potential energy it has.
• Potential energy is converted into other forms of energy like kinetic energy.
• In a pulley system, the potential energy lost by one mass is gained by another.

Kinetic Energy

Kinetic energy is the energy that an object has due to its motion. It depends on both the mass of the object and its velocity. The formula for kinetic energy is: \[ KE = \frac{1}{2} mv^2 \] where \( KE \) is the kinetic energy, \( m \) is the mass of the object, and \( v \) is its velocity.

In our scenario, when particles A and B are released from rest, they start gaining speed, converting their initial potential energy into kinetic energy.

• Since both particles start from rest, their initial kinetic energy is zero.
• As they move, their kinetic energy increases while their potential energy decreases.
• The total energy in the system remains the same due to energy conservation.

Pulley System

A pulley system consists of wheels and ropes or strings that allow objects to be lifted with less effort. In our exercise, the pulley is smooth, meaning there's no friction affecting the motion of the strings over it. The strings are also inelastic, so they do not stretch.

Here are some characteristics of pulley systems:

• They redirect the force applied, making it easier to lift heavy objects.
• In a frictionless and inelastic setup, the only forces at play are gravitational forces and tension in the string.
• The motion of one particle affects the other due to the connection via the string.

In our case, as particle B moves down, particle A moves up by the same distance. This interdependent motion is crucial to our calculation of velocity using the principle of conservation of energy.

Mass and Motion

Mass is a measure of the amount of matter in an object, and it directly affects the object's motion when forces are applied. In our example, particles A and B have masses \( m \) and \( 2m \) respectively. This difference in mass affects how they move and the resulting changes in energy.

When analyzing motion in physics:

• Heavier objects require more force to move but gain more kinetic energy.
• The mass of the objects involved determines the system's total energy distribution.
• In our problem, the greater mass of B leads to a significant change in the system's energy as it moves.

The principle of conservation of energy allows us to find the final velocity of particles A and B by equating the total initial potential energy to the total final kinetic energy. This is why understanding the mass and its effect on motion is essential for solving such problems.