Advertisements
Advertisements
प्रश्न
Test if the following equation is dimensionally correct:
\[h = \frac{2S cos\theta}{\text{ prg }},\]
where h = height, S = surface tension, ρ = density, I = moment of interia.
Advertisements
उत्तर
\[h = \frac{2S \cos \theta}{\text{ prg }}\]
Height, [h] = [L]
Surface Tension,
\[\left[ S \right] = \frac{\left[ F \right]}{\left[ L \right]} = \frac{\left[ {MLT}^{- 2} \right]}{\left[ L \right]} = \left[ {MT}^{- 2} \right]\]
Density,
\[\left[ \rho \right] = \frac{\left[ M \right]}{\left[ I \right]} = \left[ {ML}^{- 3} T^0 \right]\]
Radius, [r] = [L], [g]= [LT−2]
Now,
\[\frac{2\left[ S \right]\cos \theta}{\left[ \rho \right]\left[ r \right]\left[ g \right]} = \frac{\left[ {MT}^{- 2} \right]}{\left[ {ML}^{- 3} T^0 \right] \left[ L \right] \left[ {LT}^{- 2} \right]} = \left[ M^0 L^1 T^0 \right] = \left[ L \right]\]
Since the dimensions of both sides are the same, the equation is dimensionally correct.
APPEARS IN
संबंधित प्रश्न
“Politics is the art of the possible”. Similarly, “Science is the art of the soluble”. Explain this beautiful aphorism on the nature and practice of science.
What are the dimensions of volume of a cube of edge a.
If two quantities have same dimensions, do they represent same physical content?
A physical quantity is measured and the result is expressed as nu where u is the unit used and n is the numerical value. If the result is expressed in various units then
Find the dimensions of linear momentum .
Find the dimensions of frequency .
Find the dimensions of
(a) angular speed ω,
(b) angular acceleration α,
(c) torque τ and
(d) moment of interia I.
Some of the equations involving these quantities are \[\omega = \frac{\theta_2 - \theta_1}{t_2 - t_1}, \alpha = \frac{\omega_2 - \omega_1}{t_2 - t_1}, \tau = F . r \text{ and }I = m r^2\].
The symbols have standard meanings.
Find the dimensions of magnetic permeability \[\mu_0\]
The relevant equation are \[F = qE, F = qvB, \text{ and }B = \frac{\mu_0 I}{2 \pi a};\]
where F is force, q is charge, v is speed, I is current, and a is distance.
The height of mercury column in a barometer in a Calcutta laboratory was recorded to be 75 cm. Calculate this pressure in SI and CGS units using the following data : Specific gravity of mercury = \[13 \cdot 6\] , Density of \[\text{ water} = {10}^3 kg/ m^3 , g = 9 \cdot 8 m/ s^2\] at Calcutta. Pressure
= hpg in usual symbols.
Test if the following equation is dimensionally correct:
\[v = \sqrt{\frac{P}{\rho}},\]
where v = velocity, ρ = density, P = pressure
Can you add three unit vectors to get a unit vector? Does your answer change if two unit vectors are along the coordinate axes?
The magnitude of the vector product of two vectors \[\left| \vec{A} \right|\] and \[\left| \vec{B} \right|\] may be
(a) greater than AB
(b) equal to AB
(c) less than AB
(d) equal to zero.
Refer to figure (2 − E1). Find (a) the magnitude, (b) x and y component and (c) the angle with the X-axis of the resultant of \[\overrightarrow{OA}, \overrightarrow{BC} \text { and } \overrightarrow{DE}\].
A spy report about a suspected car reads as follows. "The car moved 2.00 km towards east, made a perpendicular left turn, ran for 500 m, made a perpendicular right turn, ran for 4.00 km and stopped". Find the displacement of the car.
Two vectors have magnitudes 2 m and 3m. The angle between them is 60°. Find (a) the scalar product of the two vectors, (b) the magnitude of their vector product.
Prove that \[\vec{A} . \left( \vec{A} \times \vec{B} \right) = 0\].
The electric current in a charging R−C circuit is given by i = i0 e−t/RC where i0, R and C are constant parameters of the circuit and t is time. Find the rate of change of current at (a) t = 0, (b) t = RC, (c) t = 10 RC.
Write the number of significant digits in (a) 1001, (b) 100.1, (c) 100.10, (d) 0.001001.
If π = 3.14, then the value of π2 is ______
