Test if the equations are dimensionally correct, a h = 2 S cosθ ρ r g b v = P ρ c V = π P r 4 t 8 η l d v = 1 2 π m g l I where h = height, S = surface tension, ρ = density, P = pressure, V = volume, η = coefficient of viscosity, v = frequency and I = moment of inertia.

a The dimensional formula for The surface tension S = M 1 L 0 T - 2 The density ρ = M 1 L - 3 T 0 The radius r = L 1 Acceleration due to gravity g = M 0 L 1 T - 2 The dimensions of the left-hand side, height h = L 1 The dimensions of the right-hand side, RHS = M 1 L 0 T - 2 M 1 L - 3 L 1 M 0 L 1 T - 2 = M 0 L 1 T 0 The dimensions of LHS and RHS are the same. Therefore, the equation is dimensionally correct. b The dimensional formula for Pressure P = M 1 L - 1 T - 2 Density ρ = M 1 L - 3 T 0 Speed v = M 0 L 1 T - 1 The dimensions of the left-hand side, LHS = v = M 0 L 1 T - 1 The dimensions of the right-hand side, RHS = M 1 L - 1 T - 2 M 1 L - 3 T 0 = M 0 L 2 T - 2 = M 0 L 1 T - 1 The dimensions of LHS and RHS are the same. Therefore, the equation is dimensionally correct. c The dimensional formula for Pressure P = M 1 L - 1 T - 2 Radius r = L 1 time t = T 1 Length l = L 1 Coefficient of viscosity η = M 1 L - 1 T - 1 Volume V = L 3 The dimensions of the left-hand side, LHS = V = M 0 L 3 T 0 The dimensions of the right-hand side, RHS = M 1 L - 1 T - 2 L 4 T M 1 L - 1 T - 1 L = M 0 L 3 T 0 The dimensions of LHS and RHS are the same. Therefore, the equation is dimensionally correct. d The dimensional formula for Mass m = M Acceleration due to gravity g = M 0 L 1 T - 2 Length l = L 1 The moment of inertia I = M 1 L 2 T 0 Frequency ν = M 0 L 0 T - 1 The dimensions of the left-hand side, LHS = M 0 L 0 T - 1 The dimensions of the right-hand side, RHS = M LT - 2 L ML 2 = T - 1 The dimensions of LHS and RHS are the same. Therefore, the equation is dimensionally correct.

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IMPORTANT

Earn 100

The power of a $100 watt$ bulb in $CGS$ unit is given by $10^{n}$ . Then find the value of $n$ .

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Important Questions on Introduction to Physics

EASY

JEE Main

IMPORTANT

CONCEPTS OF PHYSICS [VOLUME 1]>Chapter 1 - Introduction to Physics>EXERCISES>Q 12

The normal duration of

I . Sc

. The physics practical period in Indian colleges is

100 minutes

. Express this period in

microcenturies

1 microcentury = 10^{- 6} \times 100 years

. How many

microcentury

did you sleep yesterday?

MEDIUM

JEE Main

IMPORTANT

CONCEPTS OF PHYSICS [VOLUME 1]>Chapter 1 - Introduction to Physics>EXERCISES>Q 13

The surface tension of water is

72 dyne {cm}^{- 1}

. Convert it in

SI

unit.

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IMPORTANT

CONCEPTS OF PHYSICS [VOLUME 1]>Chapter 1 - Introduction to Physics>EXERCISES>Q 14

The kinetic energy

K

of a rotating body depends on its moment of inertia

I

and its angular speed

ω

. Assuming the relation to be

K = k I^{a} ω^{b}

where,

k

is a dimensionless constant, Find the value of

(a + b)

. The moment of inertia of a sphere about its diameter is

\frac{2}{5} M r^{2}

EASY

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IMPORTANT

CONCEPTS OF PHYSICS [VOLUME 1]>Chapter 1 - Introduction to Physics>EXERCISES>Q 15

The theory of relativity reveals that mass can be converted into energy. The energy

E

so obtained is proportional to certain powers of mass

m

and

c

is the speed of light. Guess a relation among the quantities using the method of dimensions.

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IMPORTANT

CONCEPTS OF PHYSICS [VOLUME 1]>Chapter 1 - Introduction to Physics>EXERCISES>Q 16

Let

I =

current through a conductor,

R =

its resistance and

V =

potential difference across its ends. According to Ohm's law, the product of two of these quantities equals the third. Obtain Ohm's law from dimensional analysis. Dimensional formulae for

R

and

V

are

[{ML}^{2} I^{- 2} T^{- 3}]

and

[{ML}^{2} T^{- 3} I^{- 1}]

, respectively.

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IMPORTANT

CONCEPTS OF PHYSICS [VOLUME 1]>Chapter 1 - Introduction to Physics>EXERCISES>Q 17

The frequency of vibration of a string depends on the length

L

between the nodes, the tension

F

in the string and its mass per unit length

m

. Guess the expression for its frequency from the dimensional analysis?

HARD

JEE Main

IMPORTANT

CONCEPTS OF PHYSICS [VOLUME 1]>Chapter 1 - Introduction to Physics>EXERCISES>Q 18

Test if the equations are dimensionally correct,

$(a) h = \frac{2 S cosθ}{ρ r g}$

$(b) v = \sqrt{\frac{P}{ρ}}$

$(c) V = \frac{π P r^{4} t}{8 η l}$

$(d) v = \frac{1}{2 π} \sqrt{\frac{m g l}{I}}$

where $h =$ height, $S =$ surface tension, $ρ =$ density, $P =$ pressure, $V =$ volume, $η =$ coefficient of viscosity, $v =$ frequency and $I =$ moment of inertia.

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CONCEPTS OF PHYSICS [VOLUME 1]>Chapter 1 - Introduction to Physics>EXERCISES>Q 19

Let,

x

and

a

stand for distance. Is

\int \frac{d x}{\sqrt{a^{2} - x^{2}}} = \frac{1}{a} \sin^{- 1} (\frac{a}{x})

dimensionally correct?

The power of a 100 watt bulb in CGS unit is given by 10n. Then find the value of n.

Important Questions on Introduction to Physics

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The power of a $100 watt$ bulb in $CGS$ unit is given by $10^{n}$ . Then find the value of $n$ .