SHM:The Calculus Picture

Q.

A body is executing simple harmonic motion with an angular frequency 2rad/s. The velocity of the body at 20 mm displacement, when the amplitude of motion is 60 mm, is

[CPMT 1999]

1. 40 mm/s

2. 60 mm/s

3. 113 mm/s

4. 120 mm/s

Q.

A body executing simple harmonic motion has a maximum acceleration equal to 24 metres/$se{c}^2$ and maximum velocity equal to 16 meters/sec. The amplitude of the simple harmonic motion is

[MP PMT 1995; DPMT 2002; RPET 2003; Pb. PET 2004]

1. 

2. 

3. 

4. 

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. $6\pi$

Q.

A particle executes simple harmonic motion between x = -A and x=+ A. The time taken for it to go from the mean position, O to $\frac{A}{2}$ is $T_1$ and to go from $\frac{A}{2}$ to A is $T_2$. Then,

1. T₁<T₂

2. T₁>T₂

3. T₁=T2

4. T₁=2T₂

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. 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 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. 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