72.
The momentum of an electron in an orbit is $$\frac{h}{\lambda }$$ where $$h$$ is a constant and $$\lambda $$ is wavelength associated with it. The nuclear magneton of electron of charge $$e$$ and mass $${m_e}$$ is given as $${\mu _n} = \frac{{eh}}{{3672\pi {m_e}}}.$$ The dimensions of $${\mu _n}$$ are $$\left( {A \to {\text{current}}} \right)$$
73.
If dimensions of critical velocity $${v_c}$$ of a liquid flowing through a tube are expressed as $$\left[ {{\eta ^x}{\rho ^y}{r^z}} \right],$$ where $$\eta ,\rho $$ and $$r$$ are the coefficient of viscosity of liquid, density of liquid and radius of the tube respectively, then the values of $$x, y$$ and $$z$$ are given by
Key Concept According to principle of homogeneity of dimension states that, a physical quantity equation will be dimensionally correct, if the dimensions of all the terms occurring on both sides of the equations are same.
Given critical velocity of liquid flowing through a tube are expressed as
$${v_c} \propto {\eta ^x}{\rho ^y}{r^z}$$
Coefficient of viscosity of liquid, $$\eta = \left[ {M{L^{ - 1}}{T^{ - 1}}} \right]$$
Density of liquid, $$\rho = \left[ {M{L^{ - 3}}} \right]$$
Radius of a tube $$r = \left[ L \right]$$
Critical velocity of liquid $${v_c} = \left[ {M{L^0}{T^{ - 1}}} \right]$$
$$\eqalign{
& \Rightarrow \left[ {{M^0}{L^1}{T^{ - 1}}} \right] = {\left[ {M{L^{ - 1}}{T^{ - 1}}} \right]^x}{\left[ {M{L^{ - 3}}} \right]^y}{\left[ L \right]^z} \cr
& \left[ {{M^0}{L^1}{T^{ - 1}}} \right] = \left[ {{M^{x + y}}{L^{ - x - 3y + z}}{T^{ - x}}} \right] \cr} $$
Comparing exponents of $$M, L$$ and $$L,$$ we get
$$\eqalign{
& x + y = 0, - x - 3y + z = 1, - x = - 1 \cr
& \Rightarrow z = - 1,x = 1,y = - 1 \cr} $$
74.
$$A,B,C$$ and $$D$$ are four different physical quantities having different dimensions. None of them is dimensionless. But we know that the equation $$AD = C\,\ln \,\left( {BD} \right)$$ holds true. Then which of the combination is not a meaningful quantity?
Dimension of $$A \ne $$ dimension of $$\left( C \right)$$
Hence $$A - C$$ is not possible.
75.
A student performs an experiment for determination of $$g\left( { = \frac{{4{\pi ^2}\ell }}{{{T^2}}}} \right).$$ The error in length $$\ell $$ is $$\Delta \ell $$ and in time $$T$$ is $$\Delta T$$ and $$n$$ is number of times the reading is taken. The measurement of $$g$$ is most accurate for-
A
$$\Delta \ell = 5\,mm,\,\,\Delta T = 0.2\,sec,\,\,n = 10$$
B
$$\Delta \ell = 5\,mm,\,\,\Delta T = 0.2\,sec,\,\,n = 20$$
C
$$\Delta \ell = 5\,mm,\,\,\Delta T = 0.1\,sec,\,\,n = 10$$
D
$$\Delta \ell = 1\,mm,\,\,\Delta T = 0.1\,sec,\,\,n = 50$$
$$\frac{{\Delta g}}{g} = \frac{{\Delta \ell }}{\ell } + 2\frac{{\Delta T}}{T}$$
$${\Delta \ell }$$ and $${\Delta T}$$ are least and number of reading are maximum in option (D), therefore the measurement of $$g$$ is most accurate with data used in this option.
76.
The dimensions of speed and velocity are
A
$$\left[ {{L^2}T} \right],\left[ {L{T^{ - 1}}} \right]$$
B
$$\left[ {L{T^{ - 1}}} \right],\left[ {L{T^{ - 2}}} \right]$$
C
$$\left[ {LT} \right],\left[ {LT} \right]$$
D
$$\left[ {L{T^{ - 1}}} \right],\left[ {L{T^{ - 1}}} \right]$$
Momentum, $$p = m × v$$
= (3.513) × (5.00)
= 17.565 $$kg\,m/s$$
= 17.6 (Rounded off to get three significant figures)
78.
Dimensions of resistance in an electrical circuit, in terms of dimension of mass $$M,$$ of length $$L,$$ of time $$T$$ and of current $$I,$$ would be
According to Ohm’s law, $$V \propto I$$
and $$V = IR$$
Resistance, $$R = \frac{{{\text{Potential difference}}}}{{{\text{Current}}}} = \frac{V}{i} = \frac{W}{{qi}}\,\left( {\because {\text{Potential difference is equal to the work done per unit charge}}} \right)$$
So, dimensions of $$R = \frac{{{\text{Dimensions of work}}}}{{{\text{Dimensions of charge}} \times {\text{Dimensions of current}}}}$$
$$ = \frac{{\left[ {M{L^2}{T^{ - 2}}} \right]}}{{\left[ {IT} \right]\left[ I \right]}} = \left[ {M{L^2}{T^{ - 3}}{I^{ - 2}}} \right]$$
$$\eqalign{
& R = \rho \frac{l}{A} \cr
& \therefore \rho = \frac{{RA}}{l} = \frac{{\Omega {m^2}}}{m} = \Omega m \cr} $$
80.
The pressure on a square plate is measured by measuring the force on the plate and length of the sides of the plate by using the formula $$P = \frac{F}{{{\ell ^2}}}.$$ If the maximum errors in the measurement of force and length are $$6\% $$ and $$3\% $$ respectively, then the maximum error in the measurement of pressure is