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Past Years NEET
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If energy (E), velocity (V) and time (T) are chosen as the fundamental quantities, the dimensional formula of surface tension will be:
Let surface tension $s = E^{a}V^{b} \cdot T^{c}$. $MLT^{-2}/L = (ML^{2}T^{-2})^{a}(L/T)^{b}(T)^{c}$ Equating the dimension of LHS and RHS. $ML^{0}T^{-2} = M^{a}L^{2a+b}T^{-2a-b+c} \Rightarrow a = 1$, $2a + b = 0$, $-2a - b + c = -2 \Rightarrow a = 1$, $b = -2$, $c = -2$ Hence, the dimensions of surface tension are $[EV^{-2}T^{-2}]$
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If dimensions of critical velocity $v_{c}$ of a liquid flowing through a tube are expressed as $[\eta^{x}\rho^{y}r^{x}]$, 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:
Applying dimensional method: $v_{c} = \eta^{x}\rho^{y}r^{z}$ $[M^{0}LT^{-1}] = [ML^{-1}T^{-1}]^{x}[ML^{-3}T^{0}]^{y}[M^{0}LT^{0}]^{z}$ Equating powers both sides $x + y = 0$; $-x = -1 \therefore x = 1$. $1 + y = 0 \therefore y = -1$. $-x - 3y + z = 1$. $-1 - 3(-1) + z = 1$. $-1 + 3 + z = 1 \therefore z = -1$
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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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A physical quantity of the dimensions of length that can be formed out of c, G and $e^{2}/(4\pi\varepsilon_{0})$ is [ c is velocity of light, G is universal constant of gravitation and e is charge]
Let dimensions of length is related as, $L = [c]^{x}[G]^{y}[e^{2}/(4\pi\varepsilon_{0})]^{z}$ $L = [LT^{-1}]^{x}[M^{-1}L^{3}T^{-2}]^{y}[ML^{3}T^{-2}]^{z}$. $[L] = [L^{x+3y+3z}M^{-y+z}T^{-x-2y-2z}]$ Comparing both sides $-y + z = 0 \Rightarrow y = z$. $x + 3y + 3z = 1$. $-x - 4z = 0 ( \because y = z)$ From (i), (ii) & (iii) $z = y = 1/2$, $x = -2$. Hence, $L = c^{-2}[G \cdot e^{2}/(4\pi\varepsilon_{0})]^{1/2}$
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In an experiment, the percentage of error occurred in the measurement of physical quantities A, B, C and D are 1%, 2%, 3% and 4% respectively. Then the maximum percentage of error in the measurement X, where $X = (A^{2}B^{1/2})/(C^{1/3}D^{3})$ will be:
Given, $X = A^{2}B^{1/2}/(C^{1/3}D^{3})$ % error, $\Delta x/x \times 100 = 2(\Delta A/A) \times 100 + (1/2)(\Delta B/B) \times 100 + (1/3)(\Delta C/C) \times 100 + 3(\Delta D/D) \times 100$ = $2 \times 1\% + (1/2) \times 2\% + (1/3) \times 3\% + 3 \times 4\% = 2\% + 1\% + 1\% + 12\% = 16\%$
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The unit of thermal conductivity is:
In steady state, the amount of heat flowing from one face to the other face in time dt is given by $dH = kA(T_{1} - T_{2})dt/\ell$ $\Rightarrow dH/dt = (kA/\ell)\Delta T$ (k = coefficient of thermal conductivity) $\therefore k = \ell dH/(Adt\Delta T)$ Unit of $k = Wm^{-1}K^{-1}$
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Dimensions of stress are:
Stress = Force/Area. Dimension of force = $[MLT^{-2}]$. Dimension of area = $[L^{2}]$ $\therefore$ Stress = $[MLT^{-2}]/[L^{2}] = [ML^{-1}T^{-2}]$
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Taking into account of the significant figures, what is the value of 9.99m - 0.0099m?
In subtraction the number of decimal places in the result should be equal to the number of decimal places of that term in the operation which contain lesser number of decimal places. $9.99 - 0.0099 = 9.9801$. As the least number of decimal places is 3. So, answer should be 9.98m.
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If E and G respectively denote energy and gravitational constant, then $E/G$ has the dimensions of
Dimensional formula of energy $E = (1/2)mv^{2}$. $[E] = [M^{1}L^{2}T^{-2}]$ Dimensional formula of gravitational constant $G = Fr^{2}/(m_{1}m_{2})$. $[G] = [M^{-1}L^{3}T^{-2}]$ From eq. (i) & (ii) $\therefore E/G = [M^{1}L^{2}T^{-2}]/[M^{-1}L^{3}T^{-2}] = [M^{2}L^{-1}T^{0}]$. Numerical constant like, $1/2$, 1,2 or $2\pi$ has no dimension.
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If force $[F]$, acceleration $[A]$ and time $[T]$ are chosen as the fundamental physical.quantities. Find the dimensions of energy.
Energy, $E \propto F^{a}A^{b}T^{c}$. $[E] = [F^{a}][A^{b}][T^{c}] \Rightarrow [ML^{2}T^{-2}] = [MLT^{-2}]^{a}[LT^{-2}]^{b}[T]^{c}$ $[ML^{2}T^{-2}] = [M^{a}L^{a+b}T^{-2a-2b+c}]$. Comparing dimensions on both sides. $a = 1$; $a + b = 2$ and $-2 = -2a - 2b + c$ $\therefore b = 1$ and $-2 = -2 - 2 + c$ or, $c = 2 \therefore$ Dimensions of energy = $[FAT^{2}]$
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The dimensions $[MLT^{-2}A^{-2}]$ belong to the:
For circular loop at centre, $B = \mu_{0}i/(2r)$. $\mu_{0} = 2Br/I$ $|\mu_{0}| = [B][r]/[I] = [MLT^{-2}A^{-1}L^{-1}][L]/[A] = MLT^{-2}A^{-2}$
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Plane angle and solid angle have:
Plane angle unit is radian, whereas solid angle unit is steradian, but they don't have any dimensions.
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Match List-I with List-II.
Column I
- A. Gravitational constant $(G)$
- B. Gravitational potential energy
- C. Gravitational potential
- D. Gravitational intensity
Column II
- i. $[L^{2}T^{2}]$
- ii. $[M^{-1}L^{3}T^{-2}]$
- iii. $[LT^{-2}]$
- iv. $[ML^{2}T^{-2}]$
Correct answer: A → ii, B → iv, C → i, D → iii
$[G] = [Fr^{2}/(m_{1}m_{2})] = MLT^{-2}L^{2}/M^{2} = M^{-1}L^{3}T^{-2}$ $[U] = [W] = ML^{2}T^{-2}$. $[V] = [U/M] = ML^{2}T^{-2}/M = L^{2}T^{-2}$. $[I] = [F/M] = LT^{-2}$
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The area of a rectangular field (in $m^{2}$ ) of length 55.3m and breadth 25m after rounding off the value of correct significant digits is:
Area = length $\times$ breadth = $55.3 \times 25$. As, Answer of multiplication is rounded off to the same number of significant figure as present in least precise term. Here, 25 has least significant figure and it is 2 so, Answer will also have 2 SF. Therefore, correct option is (c) i.e. $14 \times 10^{2}$
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The errors in the measurement which arise due to unpredictable fluctuations in temperature and voltage supply are
Random errors is also called chance error. It occurs due to parameter which are beyound the control of experimeter.
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A metal wire has mass $(0.4 \pm 0.002)g$, radius $(0.3 \pm 0.001)mm$ and length $(5 \pm 0.02)cm$. The maximum possible percentage error in the measurement of density will nearly be
Density, $\rho$ = mass/volume = $M/(\pi r^{2}l)$ $\Delta\rho/\rho = \Delta M/M + 2\Delta r/r + \Delta l/l = (0.002/0.4 + 2 \times 0.001/0.3 + 0.02/5)$ $\Delta\rho/\rho = 0.0156$. % error in density $\Delta\rho/\rho$ % = $1.56\% \approx 1.6\%$