# Question #f3570

Aug 10, 2015

Dependent on amplitude and
Conserved

#### Explanation:

Mechanical energy is the sum of Kinetic energy and Potential energy.
${E}_{o} = K + U$

Imagine an object moving in a circle with constant angular velocity, $\omega$.

Let displacement, $x = A \sin \left(\omega t\right)$
$v = \frac{\mathrm{dx}}{\mathrm{dt}} = A \omega \cos \left(\omega t\right)$

The kinetic energy,$K$ is $\frac{1}{2} m {v}^{2} = \frac{1}{2} m {A}^{2} {\omega}^{2} {\cos}^{2} \left(\omega t\right)$
Maximum kinetic energy will be accompanied by zero potential energy:
${E}_{o} = {K}_{\max} + {U}_{\min}$
$= {K}_{\max}$
Therefore, ${K}_{\max} = \frac{1}{2} m {A}^{2} {\omega}^{2} {\cos}^{2} \left(2 \pi\right) = \frac{1}{2} m {A}^{2} {\omega}^{2}$
and since ${E}_{o} = {K}_{\max}$,

${E}_{o} = \frac{1}{2} m {A}^{2} {\omega}^{2}$

From the equation above, I conclude that mechanical energy is:

• Dependent on amplitude
• Dependent on mass
• Conserved (${E}_{o}$ is just a multiplication of constants as derived above)