Physics Oscillations

The ranges and heights for two projectiles projected with the same initial velocity at angles 42° and 48° with the horizontal are R₁, R₂ and H₁, H₂ respectively. Choose the correct option:

A mass of 5kg is connected to a spring. The potential energy curve of the simple harmonic motion executed by the system is shown in the figure. A simple pendulum of length 4m has the same period of oscillation as the spring system. What is the value of acceleration due to gravity on the planet where these experiments are performed?

A bob of mass 'm' suspended by a thread of length I undergoes simple harmonic oscillations with time period T. If the bob is immersed in a liquid that has density $\frac{1}{4}$  times that of the bob and the length of the thread is increased by 1/3^rd of the original length, then the time period of the simple harmonic oscillations will be:-

For a body executing S.H.M. :

(a) Potential energy is always equal to its K.E.

(b) Average potential and kinetic energy over any given time interval are always equal.

(c) Sum of the kinetic and potential energy at any point of time is constant.

(d) Average K.E. in one time period is equal to average potential energy in one time period.

Choose the most appropriate option from the options given below:

A particle of mass 1 kg is hanging from a spring of force constant 100 Nm^-1. The mass is pulled slightly downward and released so that it executes free simple harmonic motion with time period T. The time when the kinetic energy and potential energy of the system will become equal, is $\frac{T}{x}.$  The value of x is________.

Two simple harmonic motion, are represented by the equations y₁ = 10sin $\left(3\pi t+\frac{\pi }{3}\right)$ $y_2=5\left(sin3\pi t+\sqrt[]{3}cos3\pi t\right)$  

Ratio of amplitude of y₁ to y₂ = x:1. The value of x is_______ 

A tuning fork is vibrating at 250 Hz. The length of the shortest closest organ pipe that will resonate with the tuning fork will be_____cm.

(Take speed of sound in air as 340 ms^-1)

The variation of displacement with time of a particle executing free simple harmonic moton is shown in the figure. The potential energy U(x) versus time (t) plot of the particle is correctly shown in figure:

Two simple harmonic motions are represented by the equations x₁ = 5 $sin\left(2\pi t+\frac{\pi }{4}\right)$  and $x_2=5\sqrt[]{2}\left(sin2\pi t+cos2\pi t\right).$  The amplitude of second motion is________ times the amplitude in first motion.

The acceleration due to gravity is found upto an accuracy of 4% on a planet. The energy supplied to a simple pendulum of known mass 'm' to undertake oscillations of time period T is being estimated. If time period is measured to an accuracy of 3%, the accuracy to which E is known as_______%.