Download App

Units and Measurements — Physics STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 SciencePhysicsUnits and Measurements3 Marks
Question
Consider a simple pendulum, having a bob attached to a string, that oscillates under the action of the force of gravity. Suppose that the period
of oscillation of the simple pendulum depends on its length $(l),$ mass of the bob $(m)$ and acceleration due to gravity $(g)$. Derive the expression for its time period using method of dimensions.
✓

Answer

The dependence of time period $T$ on the quantities $l, g$ and $m$ as a product may be written as :
$T=k P^r g^y m^z$
where $k$ is dimensionless constant and $x, y$ and $z$ are the exponents.
By considering dimensions on both sides, we have
${\left[ L ^0 M ^0 T ^1\right]=\left[ L ^1\right]^x\left[ L ^1 T ^{-2}\right]^y\left[ M ^1\right]^z}$
$= L ^{x \neq y} T ^{2 y} M ^z$
On equating the dimensions on both sides, we have
$x+y=0 ;-2 y=1 \text {; and } z=0$
So that $x=\frac{1}{2}, y=-\frac{1}{2}, z=0$
Then, $T=k\ l^\frac{1}{2} g^\frac{1}{2}$
or, $T=k \sqrt{\frac{l}{g}}$
Note that value of constant $k$ can not be obtained by the method of dimensions.
Here it does not matter if some number multiplies the right side of this formula,
because that does not affect its dimensions.
Actually, $k=2 \pi$ so that $T=2 \pi \sqrt{\frac{l}{g}}$

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 "3 Marks Question" from Units and Measurements→Units and Measurements — all question types→Physics STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

ls it always true that for two sources of equal intensity, the number of photons emitted in a given time are equal?
→
A motor car is travelling at $30m/ s$ on a circular road of radius $500m$. It is increasing its speed at the rate of $2ms^2$. What is the acceleration?
→
The radius of curvature of a concave mirror measured by spherometer is $\text{R}=\frac{\text{l}^2}{6\text{h}}+\frac{\text{h}}{2}$ The values of 1 and h are 4cm and 0.065cm respectively. Compute the error in measurement of radius of curvature.
→
Do all the thermocouples have a neutral temperature?
→
If a piece of paper is placed at the position of a virtual image of a strong light source, will the paper burn after sufficient time? What happens if the image is real? What happens if the image is real but the source is virtual?
→
A particle located at x = 0 at t = 0 starts moving along the positive x-direction with a velocity v that varies as $\text{v}=\alpha\sqrt{\text{x}}.$ How does the displacement of the particle vary with time?
→
Show dimensionally that the frequency n of transverse waves in a string of length / and mass, per unit length z under a tension T is given by n = $\frac{\text{K}}{l}\sqrt{\frac{\text{T}}{\text{m}}}.$
→
Consider a 10cm long piece of a wire which carries a current of 10A. Find the magnitude of the magnetic field due to the piece at a point which makes an equilateral triangle with the ends of the piece.
→
Prove that gravitational potential difference is the work done in carrying a unit mass from one point to another.
→
Find the displacement of a simple harmonic oscillator at which its PE is half of the maximum energy of the oscillator.
→