## An object of mass m is suspended at the end of a massless wire of length L and area of cross-section A ….

Q: An object of mass m is suspended at the end of a massless wire of length L and area of cross-section A . Young modulus of the material of wire is Y . If the mass is pulled down slightly its frequency of oscillation along the vertical direction is

(a) $\displaystyle f = \frac{1}{2 \pi}\sqrt{\frac{m L}{Y A}}$

(b) $\displaystyle f = \frac{1}{2 \pi}\sqrt{\frac{Y A}{m L}}$

(c) $\displaystyle f = \frac{1}{2 \pi}\sqrt{\frac{m A}{Y L}}$

(d) $\displaystyle f = \frac{1}{2 \pi}\sqrt{\frac{Y L}{m A}}$

Click to See Solution :
Ans: (b)
Sol: Frequency of oscillation $\displaystyle f = \frac{1}{2 \pi}\sqrt{\frac{k_{eq}}{m}}$

$\displaystyle Y = \frac{Stress}{Strain}$

$\displaystyle Y = \frac{F/A}{(\Delta L/L)}$

$\displaystyle F = (\frac{Y A}{L})\Delta L$

Equivalent Stiffness constant $\displaystyle k_{eq} = \frac{Y A}{L}$

$\displaystyle f = \frac{1}{2 \pi}\sqrt{\frac{Y A}{m L}}$

## When a particle of mass m is attached to a vertical spring of spring constant k and released , its motion is described ….

Q: When a particle of mass m is attached to a vertical spring of spring constant k and released , its motion is described by y(t) = yo sin2 ω t , Where ‘y’ is measured from the lower end of unstretched spring . The ω is

(a) $\displaystyle \frac{1}{2} \sqrt{\frac{g}{y_o}}$

(b) $\displaystyle \sqrt{\frac{g}{y_o}}$

(c) $\displaystyle \sqrt{\frac{g}{2 y_o}}$

(d) $\displaystyle \sqrt{\frac{2 g}{y_o}}$

Click to See Solution :
Ans: (c)

Sol: $\displaystyle y(t) = y_o sin^2 \omega t$

$\displaystyle y(t) = \frac{y_o}{2} (1-cos2 \omega t)$

Here angular frequency of oscillation ω’ = 2 ω

$\displaystyle \omega’ = \sqrt{\frac{k}{m}}$

$\displaystyle 2 \omega = \sqrt{\frac{k}{m}}$

$\displaystyle \omega = \frac{1}{2} \sqrt{\frac{k}{m}}$

Also $\large mg = k (Amplitude ) = k \frac{y_o}{2}$

$\displaystyle \frac{k}{m} = \frac{2 g}{y_o}$

$\displaystyle \omega = \sqrt{\frac{g}{2 y_o}}$

## Equation of a plane progressive wave is given by y = 0.6 sin 2π(t – x/2). On reflection from a denser medium its amplitude becomes 2/3 of the amplitude of the incident wave. The equation of the reflected wave is

Q: Equation of a plane progressive wave is given by y = 0.6 sin 2π(t – x/2). On reflection from a denser medium its amplitude becomes 2/3 of the amplitude of the incident wave. The equation of the reflected wave is

(a) y = 0.6 sin 2π(t + x/2)

(b) y = – 0.4 sin 2π(t + x/2)

(c) y = 0.4 sin 2π(t + x/2)

(d) y = – 0.4 sin 2π(t – x/2)

Ans: (b)

## A body of mass m is situated in a potential field U(x) = U_0 (1 – cos⁡αx) when, U_0 and α are constants. Find the time period of small oscillations.

Q: A body of mass m is situated in a potential field U(x) = U_0 (1 – cos⁡αx) when, U_0 and α are constants. Find the time period of small oscillations.

## A body is performing SHM, then its

Q: A body is performing SHM, then its

(a) Average total energy per cycle is equal to its maximum kinetic energy

(b) Average kinetic energy per cycle is equal to half of its maximum kinetic energy

(c) Mean velocity over a complete cycle is equal to 2/π times of it maximum velocity

(d) Root mean square velocity is 1/√2 times of its maximum velocity

Ans: (a) , (b) & (d)