Oscillations

A body is in simple harmonic motion with time period half second (T = 0.5 s) and amplitude one cm (A = 1 cm). Find the average velocity in the interval in which it moves from equilibrium position to half of its amplitude.

A solid sphere of radius $R$ and density $\rho$ is attached to one end of a mass-less spring of force constant $k$. The other end of the spring is connected to another solid sphere of radius $R$ and density $3\rho$. The complete arrangement is placed in a liquid of density $2\rho$ and is allowed to reach equilibrium. The correct statement(s) is (are)

A particle of mass $m$ is attached to one end of a mass-less spring of force constant $k$, lying on a frictionless horizontal plane. The other end of the spring is fixed. The particle starts moving horizontally from its equilibrium position at time $t=0$ with an initial velocity $u_0$. When the speed of the particle is $0.5u_0$, it collides elastically with a rigid wall. After this collision,

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 released from rest at $t=0$. It then executes simple harmonic motion with angular frequency $\omega=\dfrac\pi3$ rad/s. Simultaneously at $t=0$, a small pebble is projected with speed $v$ from point $P$ at an angle of 45° as shown in the figure. Point $P$ is at a horizontal distance of 10 m from $O$. If the pebble hits the block at $t=1$ s, the value of $v$ is (take $g=10\ \mathrm{m/s^2}$)

A metal rod of length 'L' and mass 'm' is pivoted at one end. A thin disk of mass 'M' and radius 'R' (< L) is attached at its center to the free end of the rod. Consider two ways the disc is attached: (case A) The disc is not free to rotate about its center and (case B) the disc is free to rotate about its center. The rod-disc system performs SHM in vertical plane after being released from the same displaced position. Which of the following statement(s) is (are) true?

The phase space diagram for simple harmonic motion is a circle centered at the origin. In the figure, the two circles represent the same oscillator but for different initial conditions, and $E_1$ and $E_2$ are the total mechanical energies respectively. Then

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

A point mass is subjected to two simultaneous sinusoidal displacements in x-direction, $x_1(t)=A\sin\omega t$ and $x_2(t)=A\sin\left(\omega t+\dfrac{2\pi}3\right)$. Adding a third sinusoidal displacement $x_3(t)=B\sin(\omega t+\phi)$ brings the mass to a complete rest. The values of $B$ and $\phi$ are

A wooden block performs SHM on a frictionless surface with frequency, $\nu_0$. The block carries a charge $+Q$ on its surface. If now a uniform electric field $\vec E$ is switched-on as shown, then the SHM of the block will be

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

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

A 0.1 kg mass is suspended from a wire of negligible mass. The length of the wire is 1m and its cross-sectional area is $4.9\times10^{-7}\ \mathrm m^2$. If the mass is pulled a little in the vertically downward direction and released, it performs simple harmonic motion of angular frequency 140 rad s⁻¹. If the Young's modulus of the material of the wire is $n\times10^9$ N m⁻², the value of n is

The $x$-$t$ graph of a particle undergoing simple harmonic motion is shown below. The acceleration of the particle at $t=4/3$ s is

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

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 horizontal plane. The rod is gently pushed through a small angle $\theta$ in one direction and released. The frequency of oscillation is

A block (B) is attached to two unstretched springs S1 and S2 with spring constants $k$ and $4k$, respectively (see figure I). The other ends are attached to identical supports M1 and M2 not attached to the walls. The springs and supports have negligible mass. There is no friction anywhere. The block B is displaced towards wall 1 by a small distance $x$ (figure II) and released. The block returns and moves a maximum distance $y$ towards wall 2. Displacements $x$ and $y$ are measured with respect to the equilibrium position of the block B. The ratio $\frac yx$ is

The net external force acting on the disk when its centre of mass is at displacement $x$ with respect to its equilibrium position is

The centre of mass of the disk undergoes simple harmonic motion with angular frequency $\omega$ equal to

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 II. Indicate your answer by darkening the appropriate bubbles of the 4 × 4 matrix given in the ORS.

Column I describes some situations in which a small object moves. Column II describes some characteristics of these motions. Match the situations in Column I with the characteristics in Column II and indicate your answer by darkening appropriate bubbles in the 4 × 4 matrix given in the ORS.