Work, Energy And Power

The potential energy of a point particle with mass moving along the -axis is described by: We need to determine the maximum speed attained by the particle if it starts from rest at .The particle achieves its maximum speed when its kinetic energy is at its maximum, which occurs when its potential energy is at a minimum. To find where the potential energy is minimized, we calculate the derivative of and set it to zero: Solving for gives: The second derivative, , confirms a minimum at this point.Applying the work-energy theorem, we calculate the kinetic energy when the potential energy is minimized:Using the initial and final potential energy values:Initial potential energy at : Final potential energy at : The change in kinetic energy, using the work-energy theorem: Expressing kinetic energy in terms of velocity: Solving for , the maximum speed:

Let us take : Using work energy conservation, we know that$ =R d l f=\mu m g \cos \theta d y= d x= \phi=\theta-\frac{\pi}{2}=-\left(\frac{\pi}{2}-\theta\right)$$ $

🔹 Step 1: Condition for the cart not to leave the trackAt the top of the loop (point BBB), thecentripetal forceis provided by theweightand thenormal reaction.At the limiting case (just not leaving the track):