A particle of mass m (starting at rest) moves vertically...

A particle of mass $m$ (starting at rest) moves vertically upwards from the surface of earth under an external force $\vec{F}$ which varies with height $z$ as $\vec{F}-(2-\alpha z) m \vec{g}$, where $\alpha$ is a positive constant. If $H$ is the maximum height to which particle rises. Then (a) $H-\frac{1}{\alpha}$ (b) Work done by $\vec{F}$ during motion upto $\frac{H}{2}$ is $\frac{3 m g}{2 \alpha}$ (c) $H-\frac{2}{\alpha}$ (d) Velocity of particle at $\frac{H}{2}$ is $\sqrt{\frac{g}{\alpha}}$

A particle of mass $m$ (starting at rest) moves vertically upwards from the surface of earth under an external force $\vec{F}$ which varies with height $z$ as $\vec{F}-(2-\alpha z) m \vec{g}$, where $\alpha$ is a positive constant. If $H$ is the maximum height to which particle rises. Then
(a) $H-\frac{1}{\alpha}$
(b) Work done by $\vec{F}$ during motion upto $\frac{H}{2}$ is $\frac{3 m g}{2 \alpha}$
(c) $H-\frac{2}{\alpha}$
(d) Velocity of particle at $\frac{H}{2}$ is $\sqrt{\frac{g}{\alpha}}$

Key Concepts

Potential Energy and Energy Conservation

Potential energy is the energy stored in an object due to its position in a force field, such as gravity. In problems of vertical motion, energy conservation principles relate the work done by forces to changes in both potential and kinetic energies, enabling the determination of key quantities like maximum height or speed at specific points.

Variable Force and Integration

When a force is not constant but varies with position, calculating the work done by the force requires integration. This process involves summing the infinitesimal work contributions over the path of the object, making it a vital tool for solving problems where the force function depends on variables such as height.

Work-Energy Theorem

This theorem states that the net work done on an object equals its change in kinetic energy. It is a central concept when analyzing mechanical problems, especially those involving forces that vary with position, as it allows one to equate the work done by an external force (through integration) to the changes in the object’s kinetic energy.

Kinematics in Systems with Position-Dependent Forces

Kinematics in systems influenced by variable forces involves analyzing the motion of particles under conditions where acceleration is a function of position. Understanding how to apply boundary conditions, such as zero velocity at the maximum height, is essential for linking the force law with the particle’s trajectory and determining quantities like velocity at intermediate points.

Transcript

Please give Ace some feedback