Question

The displacement of a 500 g mass, undergoing simple harmonic motion, is defined by the function :

Answer 1

To solve this problem, we have to write out the simple harmonic motion equation and then use the respective formula to solve. The maximum kinetic energy, maximum potential energy and the maximum mechanical energy is equal to 7.56J.

Simple Harmonic Motion

The given equation of the simple harmonic motion is

$x = 3.5sin(\frac{\pi}{2t} + \frac{5\pi}{4})$

Data;

• ω = π/2
• k = 1.254N/m

Solving this

$\frac{dx}{dt} = \theta = -3.5 * \frac{\pi}{2} cos (\frac{\pi }{2}t + \frac{5\pi}{4})$

Let's calculate the maximum velocity.

$V_m = \frac{3.5\pi}{2}$

This is only possible when cos θ = -1

The maximum kinetic energy is

$K_m = \frac{1}{2}mv_m^2 = \frac{1}{2}* \frac{500}{1000} * (\frac{7\pi}{4} )^2\\K_m = 7.56J$

The maximum kinetic energy is 7.56J

The maximum potential energy can be calculated as

$\omega^2 = \frac{k}{m} \\k = \omega^2 m \\k = \frac{\pi^2}{4} * \frac{500}{1000}\\ k= 1.254 N/m$

Using the value of spring constant, we can find the maximum potential energy

$P.E = \frac{1}{2}kx^2\\P.E = \frac{1}{2} * 1.234 * 3.5^2\\P.E = 7.56J$

The maximum potential energy is 7.56J

The maximum mechanical energy is equal to the sum of maximum potential energy and the maximum kinetic energy.

$ME = P.E + K.E\\ME = 7.56J$

The reason ME is equal to 7.56J is because when x is at maximum, v = 0 and makes kinetic energy equal to zero.

ME = 7.56J

From the calculations above, the maximum kinetic energy, maximum potential energy and the maximum mechanical energy is equal to 7.56J.

Learn more on simple harmonic motion here;

brainly.com/question/15556430

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