A 2.0-kg mass oscillates in one dimension with simple harmonic motion on the end of a massless spring on a horizontal frictionless table according to x = 6/ $\pi$) cos $\pi$t + 3 $\pi$) where the units are SI. The total mechanical energy of this system is
A) 1.0 J
B) 3.0 J
C) 5.0 J
D) 7.0 J
E) 9.0 J

Question

Quizplus · Accepted Answer

The total mechanical energy in simple harmonic motion is given by $ E = \frac{1}{2} k A^2 $, where $ A $ is the amplitude. From the equation $ x = \frac{6}{\pi} \cos(\pi t) + \frac{3}{\pi} $, the amplitude $ A $ is $ \frac{6}{\pi} $. The spring constant $ k $ can be found using $ \omega^2 = \frac{k}{m} $, where $ \omega = \pi $ and $ m = 2.0 \, \text{kg} $. Solving gives $ k = 2\pi^2 $. Substituting these into the energy formula gives $ E = \frac{1}{2} \times 2\pi^2 \times \left(\frac{6}{\pi}\right)^2 = 9 \, \text{J} $.

A 20-Kg Mass Oscillates in One Dimension with Simple Harmonic Motion