Simple Harmonic Motion

Two massless springs with spring constants 2 k and 9 k, carry 50 g and 100 g masses at their free ends. These two masses oscillate vertically such that their maximum velocities are equal. Then, the ratio of their respective amplitudes will be :

Assume there are two identical simple pendulum clocks. Clock - 1 is placed on the earth and Clock - 2 is placed on a space station located at a height h above the earth surface. Clock - 1 and Clock - 2 operate at time periods 4 s and 6 s respectively. Then the value of h is - (consider radius of earth $R_{E}=6400 \mathrm{~km}$ and $\mathrm{g}$ on earth $10 \mathrm{~m} / \mathrm{s}^{2}$ )

In forced oscillation of a particle the amplitude is maximum for a frequency ${\omega _1}$ of the force while the energy is maximum for a frequency ${\omega _2}$ of the force; then

If a simple pendulum has significant amplitude (up to a factor of $1/e$ of original ) only in the period between $t = 0s\,\,to\,\,t = \tau \,s,$ then $\tau \,$ may be called the average life of the pendulum When the spherical bob of the pendulum suffers a retardation (due to viscous drag) proportional to its velocity with $b$ as the constant of proportionality, the average life time of the pendulum is (assuming damping is small) in seconds :

The amplitude of a damped oscillator decreases to $0.9$ times its original magnitude in $5s$. In another $10s$ it will decrease to $\alpha$ times its original magnitude, where $\alpha$ equals

The time 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 block of mass 0.1 kg is connected to an elastic spring of spring constant 640 Nm^−1 and oscillates in a damping medium of damping constant 10^−2 kg s^−1 . The system dissipates its energy gradually. The time taken for its mechanical energy of vibration to drop to half of its initial value, is closest to :

A closed organ pipe has a fundamental frequency of 1.5 kHz. The number of overtones that can be distinctly heard by a person with this organ pipe will be (Assume that the highest frequency a person can hear is 20,000 Hz)

A simple pendulum of length 1 m is oscillating with an angular frequency 10 rad/s. The support of the pendulum starts oscillating up and down with a small angular frequency of 1 rad/s and an amplitude of 10^–2 m. The relative change in the angular frequency of the pendulum is best given by :

A damped harmonic oscillator has a frequency of 5 oscillations per second. The amplitude drops to half its value for every 10 oscillations. The time it will take to drop to 1/1000 of the original amplitude is close to :-

The displacement of a damped harmonic oscillator is given by x(t ) = e^–0.1t cos (10$\pi$t + f). Here t is in seconds. The time taken for its amplitude of vibration to drop to half of its initial value is close to :

Amplitude of a mass-spring system, which is executing simple harmonic motion decreases with time. If mass = 500g, Decay constant = 20 g/s then how much time is required for the amplitude of the system to drop to drop to half of its initial value? (ln 2 = 0.693)

A block of mass 1 kg attached to a spring is made to oscillate with an initial amplitude of 12 cm. After 2 minutes the amplitude decreases to 6 cm. Determine the value of the damping constant for this motion . (take ln 2 = 0.693)

If a spring has time period $T,$ and is cut into $n$ equal parts, then the time period of each part will be

In a simple harmonic oscillator, at the mean position

The displacement of particle varies according to the relation $x=4$$\left( {\cos \,\pi t + \sin \,\pi t} \right).$ The amplitude of the particle is

A body executes simple harmonic motion. The potential energy $(P.E),$ the kinetic energy $(K.E)$ and total energy $(T.E)$ are measured as a function of displacement $x.$ Which of the following statements is true ?

The total energy of particle, executing simple harmonic motion is

The function ${\sin ^2}\left( {\omega t} \right)$ represents