Statement 1 When a ball is gently placed on the free end of a vertically fixed spring, its kinet

Statement 1 When a ball is gently placed on the free end of...

Key Concepts

Constant versus Variable Forces

Constant forces, such as the near-uniform force of gravity, do work that is linearly related to the displacement of an object. In contrast, variable forces, like the elastic force from a spring which obeys Hooke’s law, do work that depends on the square of the displacement. Recognizing this difference is essential in understanding how work accumulates differently in systems governed by these forces.

Conservative Forces and Energy Storage

Conservative forces, such as gravity and spring (elastic) forces, are characterized by the fact that the work done by them is path independent and can be fully recovered as potential energy. This concept is key for explaining energy conservation in systems where forces like gravity and elastic forces do work.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. In dynamic systems, its changes reveal how energy is redistributed during processes like acceleration and deceleration, often due to the interplay of different forces.

Work-Energy Principle

This principle states that the total work done by all forces acting on an object is equal to the change in its kinetic energy. It is fundamental in analyzing systems where energy is transferred between different forms, such as kinetic, gravitational potential, and elastic potential energies.

Potential Energy

Potential energy is stored energy that depends on the configuration of a system. Relevant forms include gravitational potential energy, which is linked to an object’s position in a gravitational field, and elastic potential energy, which is stored when a deformable object like a spring is compressed or stretched.

Transcript

00:01 We have problem number 16 which is the assertion and region questions okay question says question says that when a ball is gently placed on a free end okay so there is a spring and this is the free end of the spring it is being told that a ball let us say of mass m gently placed so initial velocity will be equal to zero gently placed in the spring so if we assume a spring constant to be k and so it is quite natural that spring will be compressed let us say spring will be compressed by an amount x so this is the compression in the spring okay now if we draw a datum line here to calculate the potential energy gravitational potential energy so datum line here no problem let us start so a statement number one is when a ball is gently placed on okay the the it's kinetic energy first increases and then decreases to zero okay this is the statement number one second is positive work done by gravity is proportional to displacement x and work done by a spring proportional to x square and constant proportionality okay so this is statement number one and two both are interrelated it seems so let us start with the work done by gravity work done by gravity if the mass is if the ball is going in this direction so g is also in this direction so work done by gravity has to be equal has to be greater than zero so simply worked done by gravity will become equal to m g x okay so this is the work done by gravity which is m g x yes worked on proportional to we are just because when we do statement number two a statement number one could be easily proved that is which means statement number two is more relevant to a statement number one okay it will be very clear after exploring a statement number two the positive work done by gravity is proportional to yes positive work done by gravity so here work done by gravity must be positive as displacement and the gravitational force will be in the same direction okay that is vertically downward and a statement a second part of a statement two is and negative work done by spring is proportional to x square and constant okay so spring force let us say spring force is kx spring force is kx okay now this will be equal to and if we want to calculate to work done by a spring so this will be equal to kx kx is the spring force and dx will be the small displacement if we want to calculate small displacement in the direction of the spring force so this will be zero to if we integrate from 0 to x we will be getting the value okay now we can easily see that when there will be compression in a spring there will compression in…

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