A cylindrical piece of cork of density $ρ$ , base area $A$ and height $h$ floats in a liquid of density $ρ_{l}$ . The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period $T = 2 π \sqrt{\frac{h ρ}{ρ_{l} g}}$ .(Ignore damping due to viscosity of the liquid).

Important Points to Remember in Chapter -1 - Oscillations from NCERT PHYSICS PART 2 TEXTBOOK FOR CLASS XI Solutions

1. Periodic, Oscillatory and Simple Harmonic Motion:

(i) Periodic motion is that motion of a body which repeats itself after regular interval of time.

(ii) Oscillatory motion is that motion of a body during which the body moves to and fro about a fixed point after regular intervals of time.

(iii) A particle executes S.H.M. if acceleration $a \propto - y$ . Negative sign shown that acceleration $a$ is always directed in the direction opposite to that of its displacement $y$ .

2. Time period ( $T$ ) and Frequency ( $f$ ):

(i) Time Period ( $T$ ) is the time taken by a vibrating body to complete one vibration.

(ii) Frequency ( $f$ ) is defined as the number of vibrations or oscillations completed by a vibrating body in one second.

(iii) $T = \frac{1}{f}$

3. General equation of SHM:

$y = A \sin (ω t + ϕ)$

4. Amplitude:

The amplitude of a body executing S.H.M. is the maximum distance travelled by the body from its mean or equilibrium position.

5. The time period of a simple pendulum:

The time period of a simple pendulum $T = 2 π \sqrt{\frac{l}{g}}$

6. Second’s pendulum:

Second's pendulum is a pendulum whose time period is 2 seconds.

7. Time period of a spring-block system:

The time period of a spring-block system: $T = 2 π \sqrt{\frac{m}{k}}$

8. Damped oscillations:

(i) Damped Oscillations are those periodic oscillations whose amplitude decreases with the passage of time.

(ii) For small damping, amplitude can be written as $A = A_{0} e^{- b t / 2 m}$
(iii) Also for small damping, angular frequency can be written as $ω^{'} = \sqrt{\frac{k}{m} - \frac{b^{2}}{4 m^{2}}}$

9. Forced oscillations and Resonance:

(i) A system is said to execute forced oscillations when it is compelled or forced to oscillate with a frequency other than its natural frequency.

(ii) The phenomenon of producing oscillatory motion in a system by the influence of an external periodic force having the same frequency as that of the natural frequency of the system is called resonance.