A block attached to a spring undergoes simple harmonic motion on a horizontal frictionless surface.Its total energy is 50 J.When the displacement is half the amplitude, the kinetic energy is:
A)0 J
B)12.5 J
C)25 J
D)37.5 J
E)50 J

Question

Quizplus · Accepted Answer

In simple harmonic motion, the total energy (E) is the sum of potential energy (U) and kinetic energy (K). At any point, E = U + K. The potential energy in a spring system is given by $U = \frac{1}{2}kx^2$, where $x$ is the displacement from the equilibrium position. When the displacement is half the amplitude ($x = \frac{1}{2}A$), the potential energy is $U = \frac{1}{2}k\left(\frac{1}{2}A\right)^2 = \frac{1}{8}kA^2$. Since the total energy at maximum displacement (amplitude $A$) is all potential and equals $E = \frac{1}{2}kA^2 = 50 J$, when the displacement is half, the potential energy is a quarter of the total energy ($\frac{1}{4}E = 12.5 J$), leaving $50 J - 12.5 J = 37.5 J$ for the kinetic energy.

A Block Attached to a Spring Undergoes Simple Harmonic Motion