From the equation $\mathrm{tan}\theta =\frac{rg}{{v}^{2}}$, one can obtain the angle of banking θ for a cyclist taking a curve (the symbols have their usual meanings). Then say, it is

(1) Both dimensionally and numerically correct

(2) Neither numerically nor dimensionally correct

(3) Dimensionally correct only

(4) Numerically correct only

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A dimensionally consistent relation for the volume V of a liquid of coefficient of viscosity η flowing per second through a tube of radius r and length l and having a pressure difference p across its end, is

(1) $V=\frac{\pi p{r}^{4}}{8\eta l}$

(2) $V=\frac{\pi \eta l}{8p{r}^{4}}$

(3) $V=\frac{8p\eta l}{\pi {r}^{4}}$

(4) $V=\frac{\pi p\eta }{8l{r}^{4}}$

Concept Questions :-

Dimensions
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The velocity v (in cm/sec) of a particle is given in terms of time t (in sec) by the relation $v=at+\frac{b}{t+c}$; the dimensions of a, b and c are

(1) $a={L}^{2},\text{\hspace{0.17em}}b=T,\text{\hspace{0.17em}}c=L{T}^{2}$

(2) $a=L{T}^{2},\text{\hspace{0.17em}}b=LT,\text{\hspace{0.17em}}c=L$

(3) $a=L{T}^{-2},b=L,\text{\hspace{0.17em}}c=T$

(4) $a=L,\text{\hspace{0.17em}}b=LT,\text{\hspace{0.17em}}c={T}^{2}$

Concept Questions :-

Dimensions
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From the dimensional consideration, which of the following equation is correct ?

(1) $T=2\pi \sqrt{\frac{{R}^{3}}{GM}}$

(2) $T=2\pi \sqrt{\frac{GM}{{R}^{3}}}$

(3) $T=2\pi \sqrt{\frac{GM}{{R}^{2}}}$

(4) $T=2\pi \sqrt{\frac{{R}^{2}}{GM}}$

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The position of a particle at time t is given by the relation  , where v0 is a constant and α > 0. The dimensions of v0 and α are respectively

(1) ${M}^{0}{L}^{1}{T}^{-1}$ and T–1

(2) ${M}^{0}{L}^{1}{T}^{0}$ and T–1

(3) ${M}^{0}{L}^{1}{T}^{-1}$ and LT–2

(4) ${M}^{0}{L}^{1}{T}^{-1}$ and T

Concept Questions :-

Dimensions
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The dimensions of $\frac{a}{b}$ in the equation $P=\frac{a-{t}^{2}}{bx}$, where P is pressure, x is distance and t is time, are

(1) MT–2

(2) M2LT–3

(3) ML3T–1

(4) LT–3

Concept Questions :-

Dimensions
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Dimensions of $\frac{1}{{\mu }_{0}{\epsilon }_{0}}$, where symbols have their usual meaning, are

(1) [LT–1]

(2) [L–1T]

(3) [L–2T2]

(4) [L2T–2]

Concept Questions :-

Dimensions
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The dimensions of ${e}^{2}/4\pi {\epsilon }_{0}hc$, where $e,\text{\hspace{0.17em}}{\epsilon }_{0},\text{\hspace{0.17em}}h$ and c are electronic charge, electric permittivity, Planck’s constant and velocity of light in vacuum respectively

(1) $\left[{M}^{0}{L}^{0}{T}^{0}\right]$

(2) $\left[{M}^{1}{L}^{0}{T}^{0}\right]$

(3) $\left[{M}^{0}{L}^{1}{T}^{0}\right]$

(4) $\left[{M}^{0}{L}^{0}{T}^{1}\right]$

Concept Questions :-

Dimensions
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If radius of the sphere is (5.3 ± 0.1)cm. Then percentage error in its volume will be

(1) $3+6.01×\frac{100}{5.3}$

(2) $\frac{1}{3}×0.01×\frac{100}{5.3}$

(3) $\left(\frac{3×0.1}{5.3}\right)×100$

(4) $\frac{0.1}{5.3}×100$

Concept Questions :-

Errors
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The pressure on a square plate is measured by measuring the force on the plate and the length of the sides of the plate. If the maximum error in the measurement of force and length are respectively 4% and 2%, The maximum error in the measurement of pressure is

(1) 1%

(2) 2%

(3) 6%

(4) 8%

Concept Questions :-

Errors
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