Numerical Problem No. 1.8 (Measurements) Physics HSSC - I (Class -11) (Punjab Curriculum and Text Board, Lahore) With Verified Answers

 The speed v of sound waves through a medium may be assumed to depend on (a) the density 𝝆 of the medium and (b) its modulus of elasticity E which is the ratio of stress to strain. Deduce by the method of dimensions, the formula for the speed of sound.

(Ans: v = `sqrt \frac {E}{⍴}` )

Given:

Speed of sound depends on

𝝆 = density of the medium

E = Modulus of elasticity = stress : strain

To Find:

To deduce the formula for the speed of sound by the method of dimensions = v = ? 

Solution:

According to the given conditions, the formula for the speed (v) of sound will be

v ∝ `⍴^a` `E^b`

or

v = Constant `⍴^a` `E^b`  ------Equation (a)

Now using dimensions methods we will find the values of power a and b.

R.H.S. Dimensions of the equation (1):

[v] = [LT⁻¹]

L.H.S. Dimensions of the equation (1):

[⍴] = [ML⁻³]

[E] = [ML⁻¹T⁻²] 

 

(∴ E = Stress / Strain  --- Stress = Force/Area and Strain = Change in Volume / Volume)

Now writing the dimension of both sides of the equation (1) we get

[LT⁻¹] = `[ML⁻³]^a` `[ML⁻¹T⁻²]^b`

By separating each dimension quantity both side

[L] [T]⁻¹ = `[M]^{a+b}` `[L]^{-3a-b}` `[T]^{-2b}`

Now equating the power (exponent) of [M], [L], and [T] on both side, we get

By equating the powers (exponents) of  [M]

0 = a + b    ----- eqn (i)

By equating the powers  of  [L]

1 = -3a - b    ----- eqn (ii)

By equating the powers of  [T]

-1 = -2b    ----- eqn (iii)

From eqn (i) we get

b = `\frac {1}{2}` 

and substituting the value of b in eqn (i) or (ii) we get

a = -`\frac {1}{2}`

Now substituting the values of a and b in Equation (a) we have

v = Constant `⍴^-\frac {1}{2}` `E^\frac {1}{2}`

v = Constant `\frac {E^\frac {1}{2}}{⍴^\frac {1}{2}}`

v = Constant `(\frac {E}{⍴})^\frac {1}{2}`

or

v = Constant `sqrt \frac {E}{⍴}`   --------Ans

Hence, the required formula for the speed of sound

Similar Question:

Show that the expression of vf =vi +at is dimensionally correct, where vi is the velocity at t = 0 , a is acceleration and vf is the velocity at time t. 

Show that the famous "Einstein equation" E = mc² is dimensionally consistent.

Verify that the given equation S = vᵢt + `\frac {1}{2}` at² is dimensionally correct.

What are the dimensions and units of gravitational constant G in the formula 

F = G`\frac {m₁m₂}{r^2}`

Show that (a) KE = `\frac {1}{2}` mv² and  (b) PE𝖌 = mgh are dimensionally correct. 

************************************

************************************

Numerical Problem 1.8
⇑

List of 11th Physics Numerical Problems of Chapter - 1

⇑

Numerical Problem 1.7  ⇚   11th Physics Main Content List    ⇛ Numerical Problem 1.9

************************************

Shortcut Links For 

1. National MDCAT, ECAT, JOB EXAM Physics MCQs Preparation (New) 
2. Physics (New) 11th Class SLO Based Notes
3. Physics (New) 11th class MCQs Quizzes All Units
4. Physics (New) 12th Class SLO Based Notes
5. Physics (New) 12th class MCQs Quizzes All Units

1. Website for School and College Level Physics   

(https://askrarequiz.blogspot.com)

2. Website for School and College Level Mathematics  

(https://askraremaths.blogspot.com)

3. Website for Single National Curriculum Pakistan - All Subjects Notes 

(https://askrareclassnotes.blogspot.com/)

© 2022-Onwards by Academic Skills and Knowledge (ASK)     

Note:  Write me in the comment box below for any query and also Share this information with your class-fellows and friends.