SHM:The Calculus Picture

Q. A particle executes simple harmonic motion. Its amplitude is $8cm$ and time period is $6s$. The time it will take to travel from its position of maximum displacement to the point corresponding to half of its amplitude, is $s$.

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

1. 
2. 
3. 
4. 

Q.

If a simple harmonic motion is represented by $\frac{d^{2}x}{d{t}^2}+\alpha x=0$ its time period is

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

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

3. $2\pi \alpha$
   
   

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

Q. Which one of the following statements is true for the speed $v$ and the acceleration $a$ of a particle executing simple harmonic motion?

1. When $v$ is maximum, $a$ is maximum
2. Value of $a$ is zero, whatever may be the value of $v$
3. When $v$ is maximum, $a$ is zero
4. When $v$ is zero, $a$ is zero

Q. The time period of simple pendulum is $T$. If its length is increased by $2$ the new time period becomes

1. $0\cdot 99T$
2. $1\cdot 01T$
3. $1\cdot 02T$
4. $0\cdot 98T$

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. $6\pi$

3. $4\pi$

4. $2\pi$

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

Q. Two simple harmonic motions are given by $x_1=\frac{a}{2}\sin \omega t+\frac{\sqrt{3}a}{2}\cos \omega t$ and $x_2=a\sin \omega t+a\cos \omega t$. What is the ratio of the amplitude of the first motion to that of the second ?

1. $\frac{\sqrt{3}}{2}$
2. $\frac{1}{\sqrt{2}}$
3. $\sqrt{2}$
4. $\frac{1}{2}$

Q. Which one of the following statements is true for the speed $v$ and the acceleration $a$ of a particle executing simple harmonic motion?

1. When $v$ is maximum, $a$ is maximum
2. Value of $a$ is zero, whatever may be the value of $v$
3. When $v$ is zero, $a$ is zero
4. When $v$ is maximum, $a$ is zero

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. 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
2. 2
3. 2
4. 2

Q. Find the velocity when $KE=PE$ of a body undergoing SHM. Given, amplitude $=x_0$ and angular frequency is $\omega$. Also, how many times in a cycle is $KE=PE$ ?

1. $\frac{\omega x_0}{\sqrt{2}},2$
2. $\omega x_0,2$
3. $\frac{\omega x_0}{\sqrt{2}},4$
4. $\omega x_0,4$

Q. The time period of this oscillation is given by :

1. $\frac{2\pi }{\sqrt{b}}$
2. $2\pi \sqrt{b}$
3. $2\pi \sqrt{b/a}$
4. $2\pi \sqrt{a/b}$

Q. If a simple harmonic motion is represented by $\frac{d^{2}x}{d{t}^2}+\alpha x=0$. Its time period is

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

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. A string oscillates according to the equation $y^{\prime }=\left(0.50cm\right)\sin \left[\left(\frac{\pi }{3}c{m}^{-1}\right)x\right]\cos \left[\right(40\pi s^{-1}\left)t\right]$. What is the amplitude?

Q.

If a simple harmonic motion is represented by $\frac{d^{2}x}{d{t}^2}+\alpha x=0$ its time period is

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

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

3. $2\pi \alpha$
   
   

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

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. The time taken by a particle executing simple harmonic motion to move from the mean position to $(1/4{)}^{th}$ of the maximum displacement is $1\text{s}$. Calculate the angular frequency of oscillation.
$\left[{\sin }^{-1}\right(1/4)=0.252]$

1. $0.25\frac{\text{rad}}{s}$
2. $0.32\frac{\text{rad}}{s}$
3. $0.15\frac{\text{rad}}{s}$
4. $0.20\frac{\text{rad}}{s}$

Q. The displacement x (in metres) of a particle is simple harmonic motion is related to time t (in seconds) as x = $0.01cos(\pi t+\frac{\pi }{4})$ What is the frequency of the motion?

1. 0.5 Hz
2. 1.0 Hz
3. $\frac{\pi }{2}$HZ
4. $\pi HZ$

Q. A man M slides down a curved frictionless track, starting from rest. The curve obeys the equation $y=\frac{x^2}{2}$. The tangential acceleration of man is:

1. g
2. $\frac{gx}{\sqrt{x^2+4}}$
3. $\frac{g}{2}$
4. $\frac{gx}{\sqrt{x^2+1}}$

Q. A particle is subjected to three SHMs in the same direction simultaneously each having equal amplitude a and equal time period. The phase of the second motion is 30 ahead of the first and the phase of the third motion is 30 ahead of the second. Find the amplitude of the resultant motion.

Q. A particle is subjected to two simple harmonic motions of the same frequency and direction. The amplitude of the first motion is $4.0$ $cm$ and that of the second is $3.0$ $cm$. Find resultant amplitude when the phage difference is ${180}^0$. Find the resultant amplitude if the phase difference between the two motions is $0^{\circ }$.

1. $5cm$
2. $7cm$
   
3. $9cm$
   
4. $4cm$
   

Q. Two simple harmonic motion of same amplitude, same frequency and a phase difference of $\pi$ are superimposed at right angles to each other on a particle, the particle will describe a

1. circle
2. straight line
3. ellipse
4. figure of eight

Q. The time period of a particle in simple harmonic motion is equal to the smallest time between the particle acquiring a particular velocity $\overrightarrow{v}$ The value of v is

1. $v_{max}$
2. between $0$ and $v_{max}$
3. $0$
4. between $0$ and $-v_{max}$

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. Two particles execute SHM with amplitude $A$ and $2A$ and angular frequency $\omega$ and $2\omega$ respectively. At $t=0$ they starts with some initial phase difference. At $t=\frac{2\pi }{3\omega }$ they are in same phase. Their initial phase difference is

1. $\frac{2\pi }{3}$
2. $\pi$
3. $\frac{\pi }{3}$
4. $\frac{4\pi }{3}$

Q.

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

1. $2\pi \alpha$
   

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

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

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