SHM:The Calculus Picture

Q. The time period of the simple harmonic motion represented by the equation $\frac{d^{2}x}{d{t}^2}+\alpha x=0$ is

1. 2 $\pi \alpha$
2. 2 $\pi \sqrt{\alpha }$
3. 2 $\pi /\alpha$
4. 2 $\pi /\sqrt{\alpha }$

Q. From the differential equation, $\frac{d^{2}x}{d{t}^2}+16x=0$
(1) we can find time period of oscillation
(2) we cannot find time period of oscillation
(3) we cannot find initial phase of oscillation
(4) we cannot find frequency of oscillation.

1. 1 & 2
2. 2 & 3
3. 1 & 3
4. 1, 2 and 3

Q.

If an SHM is represented by $\frac{d^{2}X}{d{t}^2}+ax=0$, its time period is

1. $\frac{2\pi }{\alpha }$
   

2. $\frac{2\pi }{\sqrt{a}}$
   

3. $2\pi \alpha$
   

4. $2\pi \sqrt{\alpha }$

Q. The equation of a particle executing SHM is given by $x=150\sin (\frac{\pi }{3}t+\frac{\pi }{2})$, where the amplitude is in $\text{metres}$ and angular frequency is in $\text{rad/s}$. Find the maximum velocity $\left(v_{max}\right)$ and maximum acceleration $\left(a_{max}\right)$ attained by the particle.

1. $v_{max}=50\pi \text{m/s},a_{max}=\frac{50{\pi }^2}{3}{\text{m/s}}^2$
2. $v_{max}=100\pi \text{m/s},a_{max}=\frac{50{\pi }^2}{3}{\text{m/s}}^2$
3. $v_{max}=50\pi \text{m/s},a_{max}=\frac{100{\pi }^2}{3}{\text{m/s}}^2$
4. None of the above

Q.

If the displacement $x$ and velocity $v$ of a particle executing simple harmonic motion are related through the expression $4v^2=25-x^2$, then its time period is

1. $\pi$

2. $2\pi$

3. $4\pi$

4. $6\pi$