Oscillations Te...

A block of mass $$200$$ g executing SHM under the influence of a spring of spring constant $$k = 90 N m^{-1}$$ and a damping constant $$b = 40 g s^{-1}$$. Time taken for its amplitude of vibrations to drop to half of its initial values (Given, In $$(1/2) = -0.693)$$

A thin fixed ring of radius 1m has a positive charge $$1 \times 10^{-5}$$ C uniformly distributed over it. A particle of mass 0.9 g and having a negative charge of $$1 \times 10^-6$$ C is placed on the axis at a distance of 1 cm from the centre of the ring.  Calculate the time period of oscillations.  

A thin rod of length Land uniform area of cross-section S is pivoted at its lowest point P inside a stationary, homogenous and non-viscous liquid. The rod is free to rotate in a vertical plane about a horizontal axis passing through P. The density $$d_1$$ of the material of the rod is smaller than the density $$d_2$$ of the liquid. The rod is displaced by a small angle $$\theta$$ from its equilibrium position and then released. Show that the motion of the rod is simple harmonic and determine its angular frequency in terms of the given parameters. 

A simple harmonic oscillator of angular frequency 2 rad $$s^{-1}$$ is acted upon by an external force $$F= sin tN.$$ If the oscillator is at rest in its equilibrium position at t=0, its position at later times is proportional to 

What kind of combination of springs does this arrangement in figure shown belong to

the system shown in $$Fig. 6.330$$ is in equilibrium. Masses $${ m }_{ 1 }$$ and $${ m }_{ 2 }$$ are $$2 kg$$ and $$8 kg$$, respectively. Spring constants $${ k }_{ 1 }$$ and $${ k }_{ 1 }$$ are $$50 N{ m }^{ -1 }$$ and $$70 N{ m }^{ -1 }$$, respectively. If the compression in second spring is $$0.5 m$$. What is the compression in first spring? (Both springs have natural length initially.)

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 $$\omega =\pi /3$$ rad/s. Simultaneously at $$t=0$$, a small pebble is projected with speed v form point P at an angle of $$45^o$$ as shown in the figure. Point P is at a horizontal distance of $$10m$$ from O. If the pebble hits the block at $$t=1$$s, the value of v is (take $$g=10m/s^2$$)

A block of mass $$m$$ has initial velocity $$u$$ having direction towards '+x axis'. The block stops after covering a distance $$S$$ causing similar extension in the spring of constant $$K$$ holding it, $$\mu$$ is the kinetic friction between the block and the surface on which it was moving, the distance $$S$$ is given by: 

Three identical ideal springs, each of spring constant $$K=2000\ N/m$$ are connected in three different arrangements as shown in the figure$$-I$$, figure$$-II$$ and figure$$-III$$ respectively.
A massless hook $$A$$ is connected to the lower end of each configuration. If a mass of $$10\ kg$$ is connected to hook $$A$$, the magnitude of displacement of point $$P$$ are $${x}_{1},x_{2}$$ and $$x_{3}$$ in figure$$-I$$, Figure$$-II$$ and figure$$-III$$ respectively. Choose the correct option.

Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are charged. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along the horizontal axis and momentum is plotted along the `vertical axis. The phase space diagram is x(t) vs. p(t) curve in this plane. The arrow on the curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown in the figure. We use the sign convention in which position or momentum upwards (or to right) is positive and downwards (or to left) is negative.
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.