x = a cosωt
y = a sinωt
and z = aωt
The speed of the particle is:
A particle is executing a simple harmonic motion. Its maximum acceleration is α and maximum velocity is β. Then, its time period of vibration will be
(graph are schematic and not drawn to scale).
Here a = acceleration at time t
A uniform cylinder of length L and mass M having cross - sectional area A is suspended, with its length vertical, form a fixed point by a massless spring, such that it is half submerged in a liquid of density σ at equilibrium position. The extension x0 of the spring when it is in equilibrium is:
The amplitude of a damped oscillator decreases to 0.9 times its original magnitude in 5s. In another 10s it will decrease to α times its original magnitude, where α equals:
An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass M. The piston and the cylinder have equal cross sectional area A. When the piston is in equilibrium, the volume of the gas is V0 and its pressure if P0. The piston is slightly displaced from the equilibrium position and released. Assuming that the system is completely isolated from its surrounding, the piston executes a simple harmonic motion with frequency:
Reason The ratio of kinetic energy to potential energy is independent of the position.
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 u0. When the speed of the particle is 0.5 u0, it collides elastically with a rigid wall. After this collision,
Reason If bob is charged and kept in horizontal electric field, then the time period will be decreased.
The damping force on an oscillator is directly proportional to the velocity. The units of the constant of proportionality are :
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 E1 and E2 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 displacement in x-direction, and . Adding a third sinusoidal displacement brings the mass to a complete rest. The values of B and are
A hollow pipe of length 0.8 m is closed at one end. At its open end a 0.5 m long uniform string is vibrating in its second harmonic and it resonates with the fundamental frequency of the pipe. If the tension in the wire is 50 N and the speed of sound is 320 ms-1, the mass of the string is
Reason : Time period is directly proportional to length of pendulum.
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
Reason The periodic time depends upon the spring constant, and spring constant is large for hard spring.
The mass shown in the figure oscillates in simple harmonic motion with amplitude The amplitude of the point is
A uniform rod of length and mass is pivoted at the centre. Its two ends are attached to two springs of equal spring constant 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 in one direction and released. The frequency of oscillation is
A 20 cm long string, having a mass of 1.0 g, is fixed at both the ends. The tension in the string is 0.5 N. The string is set into vibrations using an external vibrator of frequency 100 Hz. Find the separation (in cm) between the successive nodes on the string.
x1 = vot − A(1 − cos ωt ), where A and ω are constants.
Find the position of the second block as a function of time. Also, find the relation between A and lo.