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Application of Dimensional analysis (I)- To find dimension of physical constant - (Concept)
We can find the dimension of a physical constant by substituting the dimensions of physical quantities in the given equation
-
Gravitational constant
$
\begin{aligned}
F & =G \frac{m_1 m_2}{r^2} \Rightarrow G=\frac{F r^2}{m_1 m_2} \\
G & =\frac{\left[M L T^{-2}\right]\left[L^2\right]}{[M][M]}=\left[M^{-1} L^3 T^{-2}\right]
\end{aligned}
$
$F \rightarrow$ force of Gravitation
$G \rightarrow$ Universal Gravitational Constant
$r \rightarrow$ distance between two masses
$
m_1, m_2 \rightarrow \text { two masses }
$
2. Planck's Constant(h):-
$
E=h v \Rightarrow h=\frac{E}{v}
$
Dimensional formula- $M^1 L^2 T^{-1}$
SI unit- Joule-sec
3. Rydberg constant ( R )
$
\frac{1}{\lambda}=R Z^2\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)
$
Dimension- $M^0 L^{-1} T^0$
| Exam | Chapter |
| JEE MAIN | Physics and Measurement |
Time (T), velocity (C) and angular momentum (h) are chosen as fundamental quantities instead of mass, length and time. In terms of these, the dimensions of mass would be:
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Out of the following pairs which one does not have identical dimensions is
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The speed of light (c), gravitational constant (G) and Planck's constant (h) are taken as fundamental units in a system. The dimensions of the time in the new system should be
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From the following combinations of physical constants (expressed through their usual symbols) the only combination, that would have the same value in different systems of units, is :
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Dimension formula of Rydberg constant (R) is
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If a quantity Q is given by $Q=\frac{M F \eta G}{Y T^2}$ where $M=$ mass, $F=$ force, G=gravitational constant, $\eta=$ Coefficient of viscosity, $\mathrm{Y}=\mathrm{Young}$ 's modulus. Then what is the dimension of Q in terms of $\mathrm{P}, \varepsilon_0, \mu_0$ where $\varepsilon_0=$ permittivity of free space and $\mu_0=$ permeability of free space and $\mathrm{P}=$ Power?
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A quantity $f$ is given by $f=\sqrt{\frac{h c^5}{G}}$ where c is speed on light, G is universal gravitational constant and h is the Planck's constant. Dimension of $f$ is that of:
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The amount of solar energy received on the earth's surface per unit area per unit time is defined as a solar constant. Dimension of solar constant is :
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The work done by a gas molecule in an isolated system is given by , where x is the displacement, k is the Boltzmann constant and T is the temperature,
are constants. Then the dimensions of
will be
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If 'C' and 'V' represent capacity and voltage respectively then what are the dimensions of when
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An expression for a dimensionless quantity P is given by $P=\frac{\alpha}{\beta} \log _e\left(\frac{k t}{\beta x}\right)$, where $\alpha$ and $\beta$ are constants, $\mathbf{x}$ is distance, $\mathbf{k}$ is Boltzmann constant and $\mathbf{t}$ is the temperature. Then the dimensions of $\alpha$ will be :
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The entropy of any system is given by
where are the constants.
are no. of moles, the mechanical equivalent of heat, Boltzmann constant, and gas constant respectively.
Choose the incorrect option from the following :
| A. |
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| B. |
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The force is given in terms of time and displacement
by the equation
. The dimensional formula of
is :
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| B. |
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| C. |
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| D. |
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If time , velocity
, and angular momentum
are taken as the fundamental units. Then the dimension of mass
in terms of
and
is :
| A. |
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| B. |
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| C. |
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| D. |
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The physical quantities not having same dimensions are :
| A. |
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If Energy E, velocity V and time T are taken as fundamental quantities, then the dimension formula of surface tension is :
| A. |
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| D. |
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The dimension formula of impulse is :
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Strain has same dimension as that of:
| A. |
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The dimensions of angular momentum, latent heat and capacitance are, respectively.
| A. |
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| D. |
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has the dimensions (E = electric field,
= permeability of free space)
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The dimensions of sb4 (s = Stefan's constant and b = Wein's constant) are :
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The same dimensional formula for velocity gradient is equal to :
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If mass is written as $\mathrm{m}=\mathrm{kc}^{\mathrm{P}} \mathrm{G}^{-1 / 2} \mathrm{~h}^{1 / 2}$ then the value of $\mathrm{P}$ will be : (Constants have their usual meaning with $\mathrm{k}$ a dimensionless constant)
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The equation $\left[P+\frac{a}{v^2}\right](v-b)=R T $ The unit a is
| A. |
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Given below are two statements :
Statement (I) : Dimensions of specific heat is $\left[\mathrm{L}^2 \mathrm{~T}^{-2} \mathrm{~K}^{-1}\right]$
Statement (II) : Dimensions of gas constant is $\left[\mathrm{M} \mathrm{L}^2 \mathrm{~T}^{-1} \mathrm{~K}^{-1}\right]$
| A. |
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| B. |
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| D. |
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Choose the dimensionally correct relation between $F, V, I, L, T$.
( where, $F=$ Force, $V=$ voltage, $I=$ Current $L=$ Length,$T=$ time)
| A. |
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| B. |
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Which of the them do not have same dimension formula?
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Using dimensional analysis the resistivity in terms of fundamental constants h, me, c, e, εο
can be expressed as
| A. |
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| B. |
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| D. |
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If Planck’s constant (h) and speed of light in vacuum (c) are taken as two fundamental quantities, which of the following can in addition be taken to express length, mass, and time in terms of the three chosen fundamental quantities?
a) mass of electron (me)
b) universal gravitational constant (G)
c) charge of electron (e)
d) mass of proton (mp)
| A. |
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| B. |
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| D. |
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Photon is a quantum of radiation with energy E = hv where v is the frequency and h is Planck’s constant. The dimensions of h are the same as that of:
a) linear impulse
b) angular impulse
c) linear momentum
d) angular momentum
| A. |
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| B. |
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| D. |
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In a measurement, it is asked to find modulus of elasticity per unit torque applied on the system. The measured quantity has dimension of $\left[\mathrm{M}^a \mathrm{~L}^{\mathrm{b}} \mathrm{T}^c\right]$. If $b= -3$, the value of $c$ is ________.
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If $\epsilon_0$ denotes the permittivity of free space and $\Phi_{\mathrm{E}}$ is the flux of the electric field through the area bounded by the closed surface, then dimension of $\left(\epsilon_0 \frac{\mathrm{~d} \phi_{\mathrm{E}}}{\mathrm{dt}}\right)$ are that of :
| A. |
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The position of a particle moving on $x$-axis is given by $x(t)=A \sin t+B \cos ^2 t+C t^2+D$, where $t$ is time. The dimension of $\frac{A B C}{D}$ is-
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Planck's constant (h), speed of light in vacuum (c) and Newton's gravitational constant (G) are three fundamental constants. Which of the following combinations of these has the dimension of length?
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