Physics - SHM Question with Solution | TestHub

Given $m=1$ kg, $K_{\text{max}}=8$ J, $T=4$ s, and $U_{\text{max}}=10$ J.

The minimum kinetic energy of a particle in S.H.M. is zero at the extreme positions, so $K_{\text{min}}=0$ J.

Total mechanical energy $E=U_{\text{max}}+K_{\text{min}}=10+0=10$ J.

Also, $E=K_{\text{max}}+U_{\text{min}}\Rightarrow 10=8+U_{\text{min}}\Rightarrow U_{\text{min}}=2$ J.

Angular frequency $\omega =\frac{2\pi }{T}=\frac{2\pi }{4}=\frac{\pi }{2}$ rad/s.

Maximum kinetic energy is given by $K_{\text{max}}=\frac{1}{2}m{v}_{\text{max}}^2$.

$\frac{1}{2}(1)v_{\text{max}}^2=8\Rightarrow v_{\text{max}}=4$ m/s.

Amplitude $A=\frac{v_{\text{max}}}{\omega }=\frac{4}{\pi /2}=\frac{8}{\pi }\approx 2.55$ m.

Maximum acceleration $a_{\text{max}}=\omega v_{\text{max}}=\frac{\pi }{2}\times 4=2\pi$ m/s${}^2$.

Answer: maximum acceleration of the particle is $2\pi {\text{m/s}}^2$