Oscillations

A body of mass m is attached to the lower end of a spring whose upper end is fixed; the spring has negligible mass. When the mass m is slightly pulled down and released, it oscillates with a time period of 3 s. When the mass is increased by 1 kg, the time period becomes 5 s. The value of m (in kg) is

A spring of force constant k is cut into lengths in the ratio 1 : 2 : 3. These pieces are first connected in series, giving force constant k′, and then in parallel, giving force constant k″. The ratio k′ : k″ is

A particle executes linear simple harmonic motion with an amplitude of 3 cm. When the particle is at 2 cm from the mean position, the magnitude of its velocity equals the magnitude of its acceleration. Its time period (in seconds) is

A pendulum hung from the roof of a tall building moves freely to and fro as a simple harmonic oscillator. The acceleration of the bob is 20 m/s² at a distance of 5 m from the mean position. The time period of oscillation is

A particle P revolves in a circle as shown in the figure. The radius of the circle, the period of revolution, the initial position and the sense of revolution are indicated. The y-projection of the radius vector of the rotating particle P is

The displacement of a particle executing simple harmonic motion is y = A₀ + A sin ωt + B cos ωt. The amplitude of its oscillation is

The average velocity of a particle executing SHM over one complete vibration is

The distance covered by a particle undergoing SHM in one time period is (amplitude = A)

A mass falls from a height h and its time of fall t is recorded in terms of the time period T of a simple pendulum. On the surface of the earth it is found that t = 2T. The entire set-up is taken to another planet whose mass is half that of the earth and whose radius is the same. The experiment is repeated and the corresponding times are t′ and T′. Then

The phase difference between displacement and acceleration of a particle in simple harmonic motion is

A body executes simple harmonic motion with frequency n. The frequency of variation of its potential energy is

A spring is stretched by 5 cm under a force of 10 N. The time period of oscillation when a mass of 2 kg is suspended from it is

Two pendulums of lengths 121 cm and 100 cm start swinging in phase, both at their mean position in the same phase. The minimum number of vibrations of the shorter pendulum after which the two are again in phase at the mean position is

The x–t graph of a particle performing simple harmonic motion is shown in the figure. The acceleration of the particle at t = 2 s is

A simple pendulum oscillating in air has a period of √3 s. If it is completely immersed in a non-viscous liquid whose density is one-fourth that of the material of the bob, the new period is

If x = 5 sin(πt + π/3) m represents the motion of a particle executing simple harmonic motion, then the amplitude and time period of the motion, respectively, are

If the mass of the bob of a simple pendulum is increased to thrice its original mass and its length is made half its original length, then the new time period of oscillation is (x/2) times its original time period. The value of x is

Two identical point masses P and Q, suspended from two separate massless springs of spring constants k₁ and k₂ respectively, oscillate vertically. If their maximum speeds are the same, the ratio A_Q/A_P of the amplitude of mass Q to that of mass P is

In an oscillating spring-mass system, a spring is connected to a box filled with sand. As the box oscillates, sand leaks slowly out of the box vertically, so the average angular frequency ω(t) and the average amplitude A(t) of the system change with time t. Which option correctly and schematically depicts these changes?

Consider a spring-mass simple harmonic oscillator in one dimension. The mass of the particle is $m\,$kg and the spring constant is $k\,\text{Nm}^{-1}$. At a given instant the extension of the spring is $x$ m and the speed of the particle is $v\,\text{ms}^{-1}$. On the $x$–$v$ plane, if the graph of $v$ as a function of $x$ is a circle, then the correct option is:

A cylindrical cork of uniform density floats in a liquid of density $\rho_1$. When depressed slightly and released it oscillates harmonically with time period $T$. If the same cork floats in another liquid of density $\rho_2$, the oscillation has period $2T$. The value of $\dfrac{\rho_2}{\rho_1}$ is:

For a simple pendulum having time period T, the variation of kinetic energy (K.E.) with time (t) is represented by

The sum of the kinetic energy and potential energy of a simple pendulum bob is 0.02 J. The speed of the bob at its equilibrium position is approximately (mass of the bob = 20 g)

Savitha, a Class XI student, performs an experiment to find the effective length L of a simple pendulum. She records the time for 30 oscillations as 60 s. The length she calculates is (take π² = 9.8 and g = 9.8 m/s²)