SHM expression

Q. A simple pendulum is being used to determine the value of gravitational acceleration $g$ at a certain place. The length of the pendulum is $25.0\text{cm}$ and a stop watch with $1\text{s}$ resolution measures the time taken for $40$ oscillations to be $50\text{s}$. The accuracy in $g$ is

1. $5.40\%$
2. $3.40\%$
3. $4.40\%$
4. $2.40\%$

Q. A transverse sinusoidal wave moves along a string in the positive $x-$direction at a speed of $10\text{cm/s}$. The wavelength of the wave is $0.5\text{m}$ and its amplitude is $10\text{cm}$. At a particular time $t$, the snapshot of the wave is shown in the figure. The velocity of a particle at $P$ when its displacement is $5\text{cm}$ along $y-$axis is

1. $-\frac{\sqrt{3}\pi }{50}\hat{j}\text{m/s}$
2. $\frac{\sqrt{3}\pi }{50}\hat{i}\text{m/s}$
3. $-\frac{\sqrt{3}\pi }{50}\hat{i}\text{m/s}$
4. $\frac{\sqrt{3}\pi }{50}\hat{j}\text{m/s}$

Q.

What is the dimension of angular acceleration ?

Q. In order that the resultant path on superimposing two mutually perpendicular SHM be a circle, the conditions are that:

1. the amplitudes on both SHM should be equal and they should have a phase difference of $\frac{\pi }{2}$
2. the amplitudes should be in the ratio $1:2$ and the phase difference should be zero
3. the amplitudes should be in the ratio $1:2$ and the phase difference should be $\frac{\pi }{2}$
4. the amplitudes should be equal and the phase difference should be zero

Q. A particle performs simple harmonic motion with amplitude $A$. Its speed is tripled at the instant it is at a distance $\frac{2A}{3}$ from equilibrium position. The new amplitude of the motion is

1. $\frac{A}{3}\sqrt{41}$
2. $3A$
3. $A\sqrt{3}$
4. $\frac{7A}{3}$

Q. The position of a particle is given by $r=3.0t\hat{i}-2.0t^2\hat{j}+4.0\hat{k}m$ where t is in seconds and the coefficients have the proper units for r to be in metres.

A) Find the v and a of the particle ?

B) What is the magnitude and direction of velocity of the particle at t = 2.0 s ?

Q. Two particles are executing S.H.M of the same amplitude of $20\text{cm}$ with the same period along the same line about the same equilibrium position. The maximum distance between the two is $20\text{cm}$. Their phase difference (in radians) is equal to

1. $\frac{\pi }{3}$
2. $\frac{\pi }{2}$
3. $\frac{2\pi }{3}$
4. $\frac{4\pi }{5}$

Q. A particle is placed at the lowest point of a smooth wire frame in the shape of a parabola, lying in the vertical $xy-$ plane having equation $x^2=5y$, where $(x,y)$ are in $\text{metres}$. After slight displacement, the particle is set free to move. Find the angular frequency of oscillation $\text{in rad/s}$.
[Take $g=10{\text{m/s}}^2)$]

1. $2\text{rad/s}$
2. $4\text{rad/s}$
3. $6\text{rad/s}$
4. $8\text{rad/s}$

Q. A particle performs SHM in a straight line. In the first second, starting from rest, it travels a distance $a$ and in the next second, it travels a distance $b$ on the same side of the mean position. The amplitude of the SHM is

1. $a-b$
2. $\frac{2a^2}{3a-b}$
3. None of these
4. $\frac{2a-b}{3}$

Q. If $x,v$ and $a$ denotes the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period $T$. Then which of the following does not change with time?

1. $aT/v$
2. $aT/x$
3. $a^2{T}^2+4{\pi }^2{v}^2$
4. $aT+2\pi v$

Q. The acceleration-displacement graph of a particle executing SHM is shown in given figure. The time period of its oscillation in seconds is

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

Q.

Write the dimensions of the a and b in the equation v=a+bt where v is velocity and t is time

Q. A particle of mass $1\text{kg}$ is subjected to a force which depends on the position as $\overrightarrow{F}=-k(x\hat{i}+y\hat{j})kgm{s}^{-2}$ with $k=1kg{s}^{-2}$. At time $t=0$, the particles's position $\overrightarrow{r}=(\frac{1}{\sqrt{2}}\hat{i}+\sqrt{2}\hat{j})m$ and its velocity $\overrightarrow{v}=(-\sqrt{2\hat{i}}+\sqrt{2}\hat{j}+\frac{2}{\pi }\hat{k})m{s}^{-1}$. Let $V_x$ and $V_y$ denote the $x$ and the $y$ components of the particle's velocity respectively. Ignore gravity. When $z=0.5m$, the value of $(x{v}_y-y{v}_x)$ is _________ $m^2{s}^{-1}$.

Q. A particle is moving with a velocity $\overrightarrow{v}=K(y\hat{i}+x\hat{j}),$ where $K$ is a constant. The general equation for its path is:

1. $y^2=x^2+\text{constant}$
2. $y^2=x+\text{constant}$
3. $xy=\text{constant}$
4. $y=x^2+\text{constant}$

Q. A body of mass $0.01\text{kg}$ executes simple harmonic motion (SHM) about $x=0$ under the influence of a force as shown in figure. The time period of the SHM is

1. $1.05\text{s}$
2. $0.52\text{s}$
3. $0.25\text{s}$
4. $0.31\text{s}$

Q.

The disc of mass M with uniform surface mass density $\sigma$ is shown in the figure. The center of mass of the quarter disc (the shaded area) is at the position $\frac{xa}{3\pi },\frac{xa}{3\pi }$. $x$ is ____(Round off to the Nearest Integer) [a is an area as shown in the figure]

Q.

An object of mass $m$is suspended at the end of a massless wire of length$L$ and area of cross-section$A$. Young modulus of the material of the wire is$Y.$ If the mass is pulled down slightly its frequency of oscillation along the vertical direction is:

1. $f=\left(\frac{1}{2\pi }\right)\surd \left(\frac{YA}{mL}\right)$

2. $f=\left(\frac{1}{2\pi }\right)\surd \left(\frac{mL}{YA}\right)$

3. $f=\left(\frac{1}{2\pi }\right)\surd \left(\frac{YL}{mA}\right)$

4. $f=\left(\frac{1}{2\pi }\right)\surd \left(\frac{mA}{YL}\right)$

Q.

What is rate of change of acceleration called?

Q. A particle performing $\text{S.H.M}$ is at its equilibrium when $t=1\text{s}$ and it is found to have a speed of $0.25\text{m/s}$ at $t=2\text{s}$. If the period of oscillation is $6\text{s}$. Calculate the amplitude of oscillation.

1. $\frac{3}{2\pi }\text{m}$
2. $\frac{3}{4\pi }\text{m}$
3. $\frac{6}{\pi }\text{m}$
4. $\frac{3}{8\pi }\text{m}$

Q. The time period of a simple pendulum of infinite length is ($R=$ radius of earth).

1. $\text{infinite}$
2. $2\pi \sqrt{\frac{R}{g}}$
3. $2\pi \sqrt{\frac{g}{R}}$
4. $2\pi \sqrt{\frac{2R}{g}}$

Q. The equation of motion of a particle of mass $1\text{g}$ is $\frac{d^{2}x}{d{t}^2}+{\pi }^{2}x=0$, where $x$ is displacement (in $\text{m}$) from mean position. The frequency (in $\text{Hz}$) of oscillation is

1. $0.8$
2. $0.3$
3. $0.7$
4. $0.5$

Q. A Particle is performing SHM. If its displacement from mean position is x, corresponding resultant force on it is F and acceleration is a, which of the following graphs is correct?

1. 
2. 
3. 
4. 

Q. A bead of mass $m$ stays at point $P(a,b)$ on a wire bent in the shape of a parabola $y=4C{x}^2$ and rotating with angular speed $\omega$ (see figure). The value of $\omega$ is (neglect friction)

1. $2\sqrt{2gC}$
2. $2\sqrt{gC}$
3. $\sqrt{\frac{2gC}{ab}}$
4. $\sqrt{\frac{2g}{C}}$

Q. The displacement of a particle from its mean position (in $\text{m}$) is given by $y=0.2\sin (10\pi t+1.5\pi )\cos (10\pi t+0.5\pi )$. The motion of the particle is

1. Periodic but not S.H.M
2. Non-periodic
3. S.H.M with period $0.2\text{s}$
4. S.H.M with period $0.1\text{s}$

Q. Two moving particles $P$ and $Q$ are $10\text{m}$ apart at a certain instant. The velocity of $P$ is $8\text{m/s}$ making an angle of ${30}^{\circ }$ with the line joining $P$ and $Q$. The velocity of $Q$ is $6\text{m/s}$ making an angle of ${30}^{\circ }$ with the line joining $P$ and $Q$ as shown in figure. The angular velocity of $P$ with respect to $Q$ is

1. $0\text{rad/s}$
2. $0.1\text{rad/s}$
3. $0.4\text{rad/s}$
4. $0.7\text{rad/s}$

Q. The equation of particle executing simple harmonic motion $x=10\text{sin}(\pi t+\frac{\pi }{6})$ where $x,t$ are in SI units. Find the velocity of the particle at $t=1$ seconds.

1. $5\sqrt{3}\pi {\text{ms}}^{-1}$
2. $-5\sqrt{3}\pi {\text{ms}}^{-1}$
3. $10\sqrt{3}\pi {\text{ms}}^{-1}$
4. $10\pi {\text{ms}}^{-1}$

Q.

Starting from the origin, a body oscillates simple harmonically with a period of 2 s. After what time will its kinetic energy by 75% of the total energy?

1. $\frac{1}{6}s$
   

2. $\frac{1}{4}s$
   

3. $\frac{1}{3}s$
   

4. $\frac{1}{12}s$

Q. The necessary conditions for a steady and stable interference pattern, produced by two sources of light, are,

$\left(i\right)$ two light sources must emit waves of equal frequency

$\left(ii\right)$ two point sources should have the same amplitude

$\left(iii\right)$ two sources must have constant phase difference

$\left(iv\right)$ two sources must be close to each other

1. $\left(i\right)$ and $\left(iii\right)$
2. $\left(ii\right)$ and $\left(i\right)$
3. $\left(i\right)$, $\left(ii\right)$ and $\left(iii\right)$
4. $\left(i\right)$, $\left(iii\right)$ and $\left(iv\right)$

Q. A solid ball rolls down a parabolic path $\text{ABC}$ from a height $h$ as shown in figure. Portion $\text{AB}$ of the path is rough while $\text{BC}$ is smooth. How high will the ball climb in $\text{BC}$ ?

1. 
   $\frac{5}{7}h$
2. 
   $\frac{7}{5}h$
3. 
   $\frac{2}{5}h$
4. 
   $\frac{h}{5}$

Q. A cosmic body $\text{A}$ moves to the sun with velocity $v_0$ (when far from the sun) and aimimg parmeter $l$ the arm of the vector ${\overrightarrow{v}}_0$ relative to the center of the sun. Find the minimum distance by which this body will get to the Sun. Mass of the Sun is $M$.

1. $\frac{2GM}{v_0^2}[\sqrt{1+{\left(\frac{l{v}_0^2}{GM}\right)}^2}-1]$
2. $\frac{GM}{v_0^2}$
3. $\frac{GM}{v_0^2}[\sqrt{1+\left(\frac{l{v}_0^2}{GM}\right)}-1]$
4. $\frac{GM}{2v_0^2}[\sqrt{1+{\left(\frac{l{v}_0^2}{GM}\right)}^2}-1]$