Download App

Oscillations — Physics STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 SciencePhysicsOscillations1 Mark
Question
A body of mass m is situated in a potential field $\text{U}(\text{x})=\text{U}_0(1-\cos\text{ax}),$ Where $U_0$ and a are constant. find the time period of small oscillation.
✓

Answer

$\because\text{dW}=\text{F}.\text{dx}$ if W = U, then$\text{dU}=\text{F}.\text{dx}\ \text{or}\ \text{F}=\frac{-\text{dU}}{\text{dx}}$ (here restoring force is opposite to displacement)
$\text{F}=\frac{-\text{d}}{\text{dx}}[\text{U}_0(1-\cos\text{ax})=\frac{-\text{d}}{\text{dx}}[\text{U}_0+\text{U}_0\cos\text{a}_\text{x}]$
$\text{F}=-[0-\text{U}_0(-\sin\text{ax}).\text{a}]$
$\text{F}=-\text{aU}_0\sin\text{a}\text{x}$
For SHM. ax is small
So sin ax becomes ax ...(i)
$\therefore\text{F}=-\alpha.\text{U}_0\text{ax}=-\text{a}^2\text{U}_0\text{x}\ ...(\text{ii})$
$\alpha_2\text{U}_0$ are constants.
$\therefore\text{F}\propto-\text{x}.$ so motion is SHM.
Here from (ii) $k = a^2U_0$
$\text{m}\omega^2=\text{a}^2\text{U}_0\Rightarrow\omega^2=\text{a}^2\frac{\text{U}_0}{\text{m}}$
$\Big(\frac{2\pi}{\text{T}}\Big)^2=\text{a}^2\frac{\text{U}_0}{\text{m}}\Rightarrow\text{T}^2=4\pi\frac{\text{m}}{\text{U}_0\text{a}^2}\ \text{or}$
$\text{T}=\frac{2\pi}{\text{a}}\sqrt{\frac{\text{m}}{\text{U}_0}}.$
From (i) this time period is valid for small angle ax.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "1 Marks Question" from Oscillations→Oscillations — all question types→Physics STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Why does the brake drum of an automobile get heated up while moving down at constant speed?
→
The oscillator of a pendulum is negatively charged and a positively charged conductor plate is placed below it. The pendulum is made to oscillate. What will be the effect on the time period of the pendulum?
→
A converging lens of focal length 12cm and a diverging mirror of focal length 7.5cm are placed 5.0cm apart with their principal axes coinciding. Where should an object be placed so that its image falls on itself?
→
Show graphically the temperature variation with time associated with a cooling hot body.
→
Is reversible process possible in nature?
→
Is coefficient of performance of a refrigerator constant?
→
The gravity head in a dam is 100 meters water comes out from the pipe attached to its bottom, find the velocity of water $\left( g =10 m / s ^2\right)$
→
Given below are some functions of x and t to represent the displacement (transverse or longitudinal) of an elastic wave. State which of these represent (i) a travelling wave, (ii) a stationary wave or (iii) none at all:
$\text{y}=3\sin(5\text{x}-0.5\text{t})+4\cos(5\text{x}-0.5\text{t})$
→
Resonance is an example of________.
→
A batsman hits back a ball straight in the direction of the bowler without changing its initial speed of $12 m s ^{-1}$. If the mass of the ball is $0.15 kg$, determine the impulse imparted to the ball. (Assume linear motion of the ball)
→