A bob of mass m is suspended by a string of length l from a train of mass M free to move on a smooth horizontal surface. The bob is given horizontal velocity Vo. Find the maximum height attained by the bob.

To solve the problem of a bob suspended from a train that is free to move, we need to analyze the motion of both the bob and the train. The key here is to understand the conservation of momentum and energy principles involved in the system. Let’s break it down step by step.

Understanding the System

We have a bob of mass m attached to a string of length l, and this bob is given an initial horizontal velocity V₀. The train, with mass M, is on a smooth horizontal surface, meaning there is no friction to impede its motion. When the bob is given the horizontal velocity, it will start swinging, and we need to find out how high it can rise after being set in motion.

Applying Conservation of Momentum

Initially, the bob is moving with velocity V₀ while the train is at rest. When the bob swings, it will exert a force on the train, causing the train to move. Since there are no external horizontal forces acting on the system, we can apply the conservation of momentum:

• Initial momentum of the system: p_initial = mV₀ + 0 = mV₀
• Final momentum of the system (after the bob swings and the train moves): p_final = mV + MV_t

Here, V is the final velocity of the bob at the lowest point of its swing, and V_t is the velocity of the train after the bob has been set in motion. By conservation of momentum:

mV₀ = mV + MV_t

Energy Considerations

Next, we can use the conservation of mechanical energy to find the maximum height attained by the bob. At the lowest point of the swing, all the energy is kinetic, and at the maximum height, all the energy is potential. The kinetic energy (KE) at the lowest point can be expressed as:

KE = 0.5 * m * V² + 0.5 * M * V_t²

At the maximum height h, the potential energy (PE) of the bob is:

PE = mgh

Setting the total initial kinetic energy equal to the potential energy at the maximum height gives us:

0.5 * m * V² + 0.5 * M * V_t² = mgh

Finding the Maximum Height

To find the maximum height, we need to express V and V_t in terms of V₀. From the momentum equation, we can isolate V_t:

V_t = (mV₀ - mV) / M

Substituting this into the energy equation allows us to solve for h. However, a simpler approach is to recognize that the bob will rise to a height where its kinetic energy is converted into potential energy. The maximum height can be approximated by considering the initial kinetic energy of the bob:

h = (V₀²) / (2g)

Here, g is the acceleration due to gravity. This formula gives us the maximum height attained by the bob based on its initial velocity.

Final Thoughts

In summary, the maximum height attained by the bob can be derived from the initial kinetic energy imparted to it. By applying the principles of conservation of momentum and energy, we can effectively analyze the motion of the bob and the train. The key takeaway is that the bob's height is directly related to its initial velocity and the gravitational force acting on it.