A 1.0-kg mass oscillates along the x axis with simple harmonic motion. Its position as a function of time is given by x = 2.0 cos $\pi$ t/6 + $\pi$ /3)
Where the units are SI. When t = 2.0 s, the kinetic energy of the mass is

Question

A 1.0-kg mass oscillates along the x axis with simple harmonic motion. Its position as a function of time is given by x = 2.0 cos

\pi

t/6 +

\pi

/3)
Where the units are SI. When t = 2.0 s, the kinetic energy of the mass is

Quizplus · Accepted Answer

The correct answer is A because the kinetic energy of the mass is given by:$$K = \frac{1}{2}mv^2 = \frac{1}{2}m\left(\frac{dx}{dt}ight)^2$$Substituting the given equation for x into this equation, we get:$$K = \frac{1}{2}(1.0 	ext{ kg})\left(-\frac{2\pi}{6} \sin \left(\frac{\pi}{6}t + \frac{\pi}{3}ight)ight)^2$$$$K = \frac{1}{2}(1.0 	ext{ kg})\left(\frac{\pi}{3} \sin \left(\frac{\pi}{6}t + \frac{\pi}{3}ight)ight)^2$$$$K = \frac{1}{2}\left(\frac{\pi^2}{9}ight) \sin^2 \left(\frac{\pi}{6}t + \frac{\pi}{3}ight)$$At t = 2.0 s, we have:$$K = \frac{1}{2}\left(\frac{\pi^2}{9}ight) \sin^2 \left(\frac{\pi}{6}(2.0 	ext{ s}) + \frac{\pi}{3}ight)$$$$K = \frac{1}{2}\left(\frac{\pi^2}{9}ight) \sin^2 \left(\frac{\pi}{3} + \frac{\pi}{3}ight)$$$$K = \frac{1}{2}\left(\frac{\pi^2}{9}ight) \sin^2 \left(\frac{2\pi}{3}ight)$$$$K = \frac{1}{2}\left(\frac{\pi^2}{9}ight) \left(\frac{3}{4}ight)$$$$K = \frac{\pi^2}{24} 	ext{ J}$$

A 10-Kg Mass Oscillates Along the X Axis with Simple Harmonic