Simple Harmonic Motion

A tank contains two immiscible liquids of densities $6\rho$ and $2\rho$. The higher density liquid is filled up to a height $L/2$ from the bottom. A thin rod of density $\rho$ and length $L$ is fully immersed and hinged at the bottom so tha...

As shown in the figures, a uniform rod OO' of length l is hinged at the point O and held in place vertically between two walls using two massless springs of same spring constant. The springs are connected at the midpoint and at the top-end ...

A person sitting inside an elevator performs a weighing experiment with an object of mass 50 kg . Suppose that the variation of the height $y$ (in m ) of the elevator, from the ground, with time $t$ (in s) is given by $y=8\left[1+\sin \left...

If the collision occurs at time $t_0=\pi /(2 \omega)$, then the value of $4 b^2 / a^2$ will be ______.

If the collision occurs at time $t_0=0$, the value of $v_{\mathrm{cm}} /(a \omega)$ will be ______.

On a frictionless horizontal plane, a bob of mass $m=0.1 \mathrm{~kg}$ is attached to a spring with natural length $l_{0}=0.1 \mathrm{~m}$. The spring constant is $k_{1}=0.009 \,\mathrm{Nm}^{-1}$ when the length of the spring $l>l_{0}$ and ...

A particle of mass $1 \mathrm{~kg}$ is subjected to a force which depends on the position as $\vec{F}=$ $-k(x \hat{\imath}+y \hat{\jmath}) \mathrm{kg}\, \mathrm{m} \mathrm{s}^{-2}$ with $k=1 \mathrm{~kg} \mathrm{~s}^{-2}$. At time $t=0$, th...

A block with mass M is connected by a massless spring with stiffness constant k to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude A about an equilibrium position x0. Consider two c...

Two independent harmonic oscillators of equal masses are oscillating about the origin with angular frequencies \(\omega\)1 and \(\omega\)2 and have total energies E1 and E2, respectively. The variations of their momenta p with positions x a...

A small block is connected to one end of a massless spring of un-stretched length 4.9 m. The other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless surface. The block is stretched by 0.2 m and releas...

A point mass is subjected to two simultaneous sinusoidal displacements in x-direction, \({x_1}\left( t \right) = A\sin \omega t\) and \({x_2}\left( t \right) = A\sin \left( {\omega t + {{2\pi } \over 3}} \right)\). Adding a third sinusoidal...

A wooden block performs \(SHM\) on a frictionless surface with frequency, \({v_0}.\) The block carries a charge \(+Q\) on its surface . If now a uniform electric field \(\overrightarrow E\) is switched- on as shown, then the \(SHM\) of th...

Consider the spring-mass system, with the mass submerged in water, as shown in the figure. The phase space diagram for one cycle of this system is

The phase space diagram for simple harmonic motion is a circle centred at the origin. In the figure, the two circles represent the same oscillator but for different initial conditions, and E1 and E2 are the total mechanical energies respect...

The phase space diagram for a ball thrown vertically up from ground is

A 0.1 kg mass is suspended from a wire of negligible mass. The length of the wire is 1 m and its crosssectional area is 4.9 \(\times\) 10-7 m2. If the mass is pulled a little in the vertically downward direction and released, it performs ...

The acceleration of this particle for \(|x| > {X_0}\) is

For periodic motion of small amplitude A, the time period T of this particle is proportional to

If the total energy of the particle is E, it will perform periodic motion only if

A student performed the experiment to measure the speed of sound in air using resonance air-column method. Two resonances in the air-column were obtained by lowering the water level. The resonance with the shorter air-column is the first re...

A uniform rod of length L and mass M is pivoted at the centre. Its two ends are attached to two springs of equal spring constants \(k\). The springs are fixed to rigid supports as shown in the figure, and the rod is free to oscillate in the...

The mass M shown in the figure below oscillates in simple harmonic motion with amplitude A. The amplitude of the point P is

The \(x\)-\(t\) graph of a particle undergoing simple harmonic motion is shown in the figure. The acceleration of the particle at \(t=4/3\) s is

Column I gives a list of possible set of parameters measured in some experiments. The variations of the parameters in the form of graphs are shown in Column II. Match the set of parameters given in Column I with the graphs given in Column I...

Function $x=\mathrm{A} \sin ^2 \omega t+\mathrm{B} \cos ^2 \omega t+\mathrm{C} \sin \omega t \cos \omega t$ represents SHM

A simple pendulum has time period T1. The point of suspension is now moved upward according to the relation y = Kt2, (K = 1 m/s2) where y is the vertical displacement. The time period now become T2. The ratio of \({{T_1^2} \over {T_2^2}}\) ...

A small body attached to one end of a vertically hanging spring is performing SHM about its mean position with angular frequency \(\omega\) and amplitude \(a\). If at a height \(y^{\prime}\) from the mean position, the body gets detached fr...

A particle executes simple harmonic motion between x = - A to x = + A. The time taken for it to go from 0 to \({A \over 2}\) is T1 and to go from \({A \over 2}\) to A is T2. Then

The period of oscillation of a simple pendulum of length \(L\) suspended from the roof of a vehicle which moves without friction down an inclined plane of inclination \(\alpha\), is given by

A particle free to move along the x-axis has potential energy given by \(U\left( x \right) = k\left[ {1 - \exp \left( { - {x^2}} \right)} \right]\) for \(- \infty \le x \le - \infty\), where k is a positive constant of appropriate dimen...

An object of mass 0.2 kg executes simple harmonic oscillation along the x-axis with a frequency of \(\left( {{{25} \over \pi }} \right)\) Hz. At the position x = 0.04, the object has kinetic energy of 0.5 J and potential energy 0.4 J. The a...

Two bodies M and N of equal masses are suspended from two separate massless springs of spring constant k1 and k2 respectively. If the two bodies oscillate vertically such that their maximum velocities are equal, the ratio of the amplitude o...