A block of mass is attached to a horizontal spring of force constant and executes…

PHYSICS · Multi-part · CBSE · Physics · Oscillations · Oscillations

Problem

A block of mass

0.25\,\text{kg}

is attached to a horizontal spring of force constant

100\,\text{N}\,\text{m}^{-1}

and executes simple harmonic motion (SHM) along the

x

–axis with amplitude

0.10\,\text{m}

. Answer the following: (i) Derive an expression for the potential energy

U(x)

of the block as a function of its instantaneous displacement

x

from the equilibrium position. (ii) Derive an expression for the kinetic energy

K(x)

of the block as a function of

x

. (iii) Using the expressions obtained in (i) and (ii), show that the graph of

U

versus

x

is a parabola opening upward while that of

K

versus

x

is a parabola opening downward. Give the coordinates of the intercepts and the vertex for each curve. (iv) On one set of axes, sketch the curves of

U(x)

and

K(x)

together with the straight line representing the total mechanical energy of the system. Label all significant points clearly.