The position of a particle at time t is given by the relation x t = v 0 α 1 - e - α t , v 0 where is a constant and α > 0 . Find the dimensions of v 0 and α .

From the principle of dimensional homogeneity, dimensional quantities can add or subtract if they're having same dimension. 1 is dimensionless, so [ αt ] is also dimensionless ∴ [ α ] = 1 t = T - 1 Also, left-hand side of equation should have same dimension as right-hand side of equation. [ x ] = v 0 α ∴ v 0 = [ x ] [ α ] = [ L ] [ T - 1 ] = LT - 1

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The dimensionless quantity

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Important Questions on Physical World and Measurement

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Beta Question Bank for Engineering: Physics>Chapter 3 - Physical World and Measurement>Exercise-2>Q 6

Bernoulli’s equation is given by

p + \frac{1}{2} {ρv}^{2} + hρg = k

Where

p =

pressure,

ρ =

density,

v =

speed,

h =

height of the liquid column,

g =

acceleration due to gravity and

k

is constant. The dimensional formula for

k

is same as that for :

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Beta Question Bank for Engineering: Physics>Chapter 3 - Physical World and Measurement>Exercise-2>Q 7

When a wave travels in a medium, the displacement of a particle located at distance

x

at time

t

is given by

y = a \sin (b t - c x)

where

a, b

and

c

are constants of the wave. The dimensions of

\frac{b}{c}

are the same as that of

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Beta Question Bank for Engineering: Physics>Chapter 3 - Physical World and Measurement>Exercise-3>Q 15

A projectile fired at an angle of

45 °

travels a total distance

R

, called the range, which depends only on the initial speed

v

and the acceleration of gravity

g

. Using dimensional analysis, if

R

depends on the speed and on

g

R = K \frac{v^{p}}{g^{q}}

, then the value of

p + q

is:

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Beta Question Bank for Engineering: Physics>Chapter 3 - Physical World and Measurement>Exercise-4>Q 1

The position of a particle at time

t

is given by the relation

x (t) = (\frac{v_{0}}{α}) (1 - e^{- α t})

v_{0}

where is a constant and

α > 0

. Find the dimensions of

v_{0}

and

α

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Beta Question Bank for Engineering: Physics>Chapter 3 - Physical World and Measurement>Exercise-4>Q 2

The related equation are: $Q = mc (T_{2} - T_{1}), ℓ_{1} = ℓ_{0} [1 + α (T_{2} - T_{1})]$ and $PV = nRT$ , where the symbols have their usual meanings. Find the dimensions of

(A) specific heat capacity $(C)$

(B) coefficient of linear expansion $(α)$ and

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Beta Question Bank for Engineering: Physics>Chapter 3 - Physical World and Measurement>Exercise-4>Q 4

In the formula,

p = \frac{n R T}{V - b} e^{\frac{a}{R T V}},

find the dimensions of

a

and

b,

where

p =

pressure,

n =

number of moles,

T =

temperature,

V =

volume and

R =

universal gas constant.

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Beta Question Bank for Engineering: Physics>Chapter 3 - Physical World and Measurement>Exercise-4>Q 5

If instead of mass, length and time as fundamental quantities we choose velocity, acceleration and force as fundamental quantities and express their dimensional symbols as

v, a

and

F

respectively. Show that the dimensions of Young's modulus can be expressed as

[{Fa}^{2} v^{- 4}]

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Beta Question Bank for Engineering: Physics>Chapter 3 - Physical World and Measurement>Exercise-4>Q 6

If energy

(E)

, velocity

(v)

and time

(T)

are fundamental units then the dimension of surface tension is

[{Ev}^{- α} T^{- β}]

. Find

α - β

The dimensionless quantity

Important Questions on Physical World and Measurement

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