A student plots a graph for a particle executing simple harmonic motion. The resulting graph is a straight line with a negative slope and a positive intercept on the vertical axis. Which of the following pairs of physical quantities could be plotted on the vertical ( y ) and horizontal ( x ) axes, respectively?

Step-by-step solution

For a particle in SHM, the kinetic energy ( K ), potential energy ( U ), and total energy ( E ) as functions of position x are given by: K = 1 2 m ^2(A^2 - x^2) = - 1 2 m ^2 x^2 + 1 2 m ^2 A^2 U = 1 2 m ^2 x^2 E = 1 2 m ^2 A^2 If we plot K on the y -axis and x^2 on the horizontal axis (let X = x^2 ), the equation becomes: y = - ( 1 2 m ^2 )X + 1 2 m ^2 A^2 This is the equation of a straight line ( y = -mX + c ) with a negative slope ( - 1 2 m ^2 ) and a positive y -intercept ( 1 2 m ^2 A^2 ). Plotting U vs x^2 give