Oscillations Te...

The position vector of a particle that is moving in space is given by $$\vec {r} =(1 +2\cos 2\omega t)\hat {i} + (3\sin^{2} \omega t)\hat {j} + (3)\hat {k}$$ in the ground frame. All units are in $$SI$$. Choose the correct statement(s).

The string of a pendulum is horizontal initially. The mass of the bob attached is m. Now the string is released. 'R' is the length of the pendulum.
(i) The tension in the string at an angle $$30 ^\circ$$ with the vertical is ______

A body is projected with an initial velocity $$20 \ m/s$$ at $$60^\circ$$ to the horizontal. The displacement after $$2 s$$ is:

A simple pendulum with a brass bob has a time period T. The bob is now immersed in a non-viscous liquid and made to oscillate. The density of the liquid is (1/8)th that of the brass. The time period of the pendulum will  be 

If a body of mass $$36gm$$ moves with S.H.M of amplitude $$A=13cm$$ and $$T= 12\sec$$. At time $$t=0$$ the displacement is $$x=+12cm$$. The shortest time of passage from $$x=+6.5cm$$ to $$x=-6.5$$ is 

A book is resting on shelf that is undergoing vertical simple harmonic oscillations with an amplitude of $$2.5 cm $$ . What is the minimum frequency of oscillations of the shell for which the book will the lose contact with the shelf?
(Assume that $$ g =  m/s^2 $$ )

Two pendulums of lengths 121 cm and 100 cm start vibrating at the same instant. They are in the mean position and in the same phase. After how many vibrations of the shorter pendulum, the two will be in the same phase in the mean position? 

A particle performs harmonic oscillations along a straight line with a period of 6 s and amplitude 4 cm. The mean velocity of the particle averaged over the time interval during which it travel distance of 2 cm starting from the extreme position in cm /s is

The motion of a particle is described by $$9\dfrac{d^{2}x}{dt^{2}}+25\ x=80$$ where $$x$$ is displacement and $$t$$ is time. Angular frequency of small oscillations is

A particle executes SHM with amplitude $$0.5\ cm$$ and frequency $${100s}^{-1}$$. The maximum speed of the particle is (in m/s)