The potential energy of a simple harmonic oscillator when the particle is half way to its end point is (where E is the total energy)

1.  $\frac{1}{8}E$       

2.  $\frac{1}{4}E$

3.  $\frac{1}{2}E$       

4.  $\frac{2}{3}E$

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 ?

1. P.E. is maximum when x = 0

2. K.E. is maximum when x = 0

3. T.E. is zero when x = 0

4. K.E. is maximum when x is maximum

­­A man measures the period of a simple pendulum inside a stationary lift and finds it to be T sec. If the lift accelerates upwards with an acceleration $\raisebox{1ex}{$\mathrm{g}$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$ , then the period of the pendulum will be

1. T

2. $\frac{\mathrm{T}}{4}$

3. $\frac{2\mathrm{T}}{\sqrt{5}}$

4. $2\mathrm{T}\sqrt{5}$

The total energy of a particle, executing simple harmonic motion is

1. $\propto x$                 

2. $\propto x^2$

3. Independent of x 

4.$\propto x^{1/2}$

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

1. $\sqrt{2\mathrm{gl}(1-\mathrm{sin\theta })}$

2. $\sqrt{2\mathrm{gl}(1+\mathrm{cos\theta })}$

3. $\sqrt{2\mathrm{gl}(1-\mathrm{cos\theta })}$

4. $\sqrt{2\mathrm{gl}(1+\mathrm{sin\theta })}$

A body is executing Simple Harmonic Motion. At a displacement x its potential energy is $E_1$ and at a displacement y its potential energy is $E_2$. The potential energy E at displacement $\left(x+y\right)$ is 

1.  $E=\sqrt{E_1}+\sqrt{E_2}$   

2.  $\sqrt{E}=\sqrt{E_1}+\sqrt{E_2}$

3.   $E=E_1+E_2$           

4.  None of these.

The equation of motion of a particle is $\frac{d^{2}y}{d{t}^2}+Ky=0$ where K is positive constant. The time period of the motion is given by

1. $\frac{2\mathrm{\pi }}{K}$             

2. $2\mathrm{\pi K}$

3. $\frac{2\mathrm{\pi }}{\sqrt{K}}$           

4.  $2\pi \sqrt{K}$

The kinetic energy of a particle executing S.H.M. is 16 J when it is in its mean position. If the amplitude of oscillations is 25 cm and the mass of the particle is 5.12 kg, the time period of its oscillation is -

(1) $\frac{\mathrm{\pi }}{5}sec$       

(2) $2\mathrm{\pi } \sec$

(3) $20\mathrm{\pi } \sec$   

(4) $5\mathrm{\pi } \sec$