Oscillations practice · Mixed difficulty

The average life of an Indian is 56 years. Find the number of times the human heart beats in the life of an Indian, if the heart beats once in $0.8 \mathrm{~s}$.

Choose the correct statement in the following statements.

Which one of the following is NOT a characteristic of SHM?

If the displacement equation of a particle be represented by $y = A\sin PT + B\cos PT,$ the particle executes

The bob of a simple pendulum is a hollow
sphere filled with water. If a hole is made at its
bottom, so that water emerges out slowly, then
time period of oscillations

What happens to the natural frequency of vibration of stretched spring, when its length and diameter are increased?

Out of the following functions representing motion of a particle which represents SHM (i) $y = \sin \omega t - \cos \omega t$ (ii) $y = {\sin ^3}\omega t$ (iii) $y = 5\cos \left( {\frac{{3\pi }}{4} - 3\omega t} \right)$ (iv) $y = 1 + \omega t + {\omega ^2}{t^2}$

Two pendulums of lengths 1m and 1.21m respectively start swinging together with same amplitude. The number of vibrations that will be executed by the longer pendulum before the two will swing together again are

The speed of the particle on a reference circle of radius $R$ is $V$ . The time period of oscillations of the projection on a diameter of the circle is

A spring is stretched by $0.40 \mathrm{~m}$ when a mass of $0.6 \mathrm{~kg}$ is suspended from it. The period of oscillations of the spring loaded by $255 \mathrm{~gm}$ and put to oscillations is close to__ $\left(\mathrm{g}=10 \mathrm{~m}.\mathrm{s}^{-2}\right)$

The acceleration of a particle moving along $x$-axis is $a=-200 x+50$. It is released from $x=2$. Here ' $a$ ' and ' $x$ ' are in S.I. units. The motion of particle will be :-

The Displacement of a particle varies with time as $x=12 \sin \omega t-16 \sin ^3 \omega t$ (in $\mathrm{cm}$). if its motion is S.H.M, Then its maximum accelaration is

The maximum force acting on a particle executing simple harmonic motion is $10 \mathrm{~N}$. The force on the particle when it is midway between mean and extreme positions will be

Assertion: Time average KE and time average PE are not exactly equal in simple pendulum.
Reason: Friction is not negligible in simple pendulum.

Assertion: In simple harmonic motion, the motion is to and fro and periodic
Reason: Velocity of the particle in simple harmonic motion $(v)=\omega \sqrt{{k}^2-{x}^2}$ (where ${x}$ is the displacement and ${k}$ is spring factor).

The bob of a pendulum of length l is pulled aside from its equilibrium position through an angle θ and then released. The bob will then pass through its equilibrium position with a speed $v$, where $v$ equals

Which of the following quantity is always negative in S.H.M? (Symbols have their usual meanings)

Statement-1: A particle is moving along $x$-axis. The resultant force $F$ acting on it is given by $F=-a x-b$ where $a$ and $b$ are both positive constants. The motion of this particle is not S.H.M.
Statement-2: In S.H.M. resultant force must be proportional to the displacement from mean position.

The phase difference (in radian) between two SHM whose equations are $y_1=10 \sin \left(10 \pi t+\frac{\pi}{3}\right)$ and $y_2=12 \sin \left(8 \pi t+\frac{\pi}{4}\right)$ at $t=0.5 \mathrm{~s}$ is:

The acceleration of a particle in SHM at 5 cm from its mean position is $80 \mathrm{~cm} / \mathrm{sec}^2.$ The value of angular frequency in radian/sec will be:

Which of the following functions represents a simple harmonic oscillation?

A heavy brass sphere is hung from a spring and it executes vertical vibrations with period $T$. The sphere is now immersed in a non-viscous liquid with a density $(1 / 10)$ th that of brass. When set into vertical vibrations with the sphere remaining inside liquid all the time, the time period will be

The ratio of angular speed of a second- hand to the hour-hand of a watch is:

(A): Since and cosine functions are periodic functions
(R): Sinusoidal functions repeat its values after a definite interval of time

A particle performs linear S.H.M. At a particular instant, velocity of the particle is ' $u$ ' and acceleration is ' $\alpha$ ' while at another instant velocity is ' $v$ ' and acceleration is ' $\beta$ ' $(0<\alpha<\beta)$. The distance between the two positions is

The ratio of the magnitudes of maximum velocity to the maximum acceleration of a particle making $SHM$ is $1:1$. The minimum time taken by the particle to move between the extreme points of its path is

Assertion (A): Damped vibrations indicate loss of energy.
Reason (R): The loss may be due to friction, air resistance etc.

A vertical spring with a load of mass $100\,grams$ has the same frequency of oscillation as that of a seconds pendulum at a place. The spring constant of the spring is nearly $\left[ {in\,\,N/m} \right]$

Time period of a simple pendulum of length $\ell$ is $T_1$ and time period of uniform rod of same length $\ell$ pivoted about one end and oscillating in a vertical plane is $T_2$. Amplitude of oscillations in both the cases is small, then $\frac{T_1}{T_2}$ is

A particle is executing SHM of amplitude ' $A$ ' and time period ' $T$ '. Find the time (T) taken by the particle to go from $x=0$ to $x=A / 2$.