The mechanical quantity, which has dimensions of reciprocal of mass \( \text{M}^{-1} \) is
The dimension of the universal gravitational constant \(G\) is \([\text{M}^{-1}\text{L}^3\text{T}^{-2}]\). Therefore, its dimensions contain the reciprocal of mass.
The diameter of a spherical bob, when measured with vernier callipers yielded the following values: 3.33 cm, 3.32 cm, 3.34 cm, 3.33 cm and 3.32 cm. The mean diameter to appropriate significant figures is:
The mean of the values is \(\frac{3.33 + 3.32 + 3.34 + 3.33 + 3.32}{5} = 3.328\text{ cm}\). Rounding to the least number of decimal places in the readings (two decimal places) yields 3.33 cm.
Which of the following statement is not true?
Pressure is defined as thrust per unit area. Since thrust is a normal force component, it is always perpendicular to the surface. Pressure does not have a unique direction associated with it in space, making it a scalar quantity.
Dimensional formula for torque is
Torque is defined as force multiplied by perpendicular distance: \(\tau = F \times r\). Its dimensions are \([MLT^{-2}] \times [L] = [ML^2T^{-2}]\).
Find dimensions of constants \(a\) & \(b\) in given equation: \(v = at^2\cos t + \frac{b}{t\sin\theta}\), where \(v \rightarrow\) velocity, \(t \rightarrow\) time.
By the principle of homogeneity, each term on the right must have the dimensions of velocity \(v\). Thus, \([at^2] = [v] ⇒ [a] = [LT^{-3}]\), and \([\frac{b}{t}] = [v] ⇒ [b] = [L]\).
In an experiment \(Z\) is measured as \(Z = \frac{A^{1/3} B^2}{\sqrt{C}}\). Relative error in given quantities \(A\), \(B\), & \(C\) are 0.3, 0.2 & 0.6 respectively. Find maximum relative error in \(Z\).
Using error propagation: \(\frac{\Delta Z}{Z} = \frac{1}{3}\frac{\Delta A}{A} + 2\frac{\Delta B}{B} + \frac{1}{2}\frac{\Delta C}{C}\). Substituting values: \(\frac{1}{3}(0.3) + 2(0.2) + \frac{1}{2}(0.6) = 0.1 + 0.4 + 0.3 = 0.8\).
Using rules of significant figures what is the value of \(22.8\text{ m} \times 3.4\text{ m}\):
The multiplication result must be rounded to the least number of significant figures among the measured values. Since 3.4 has only 2 significant figures, the result \(77.52\text{ m}^2\) is rounded to 2 significant figures, which is \(78\text{ m}^2\).
Weight measured by a spring balance gives following readings: 40 N, 42 N, 44 N, 39 N, 45 N. What is the mean absolute error of the observations?
Mean value is \(42\text{ N}\). Absolute errors are \(|40-42|=2\), \(|42-42|=0\), \(|44-42|=2\), \(|39-42|=3\), \(|45-42|=3\). Mean absolute error is \(\frac{2+0+2+3+3}{5} = 2\text{ N}\).
Assertion: Mass, length and time may be taken as fundamental quantities.
Reason: Mass, length and time are independent of one another.
Mass, length, and time are chosen as fundamental quantities because they cannot be defined in terms of each other and are completely independent.
Match the column:\n(a) Pressure -> (i) \([ML^2T^{-1}]\)\n(b) Angular Momentum -> (ii) \([M^{-1}L^{-3}T^4A^2]\)\n(c) Magnetic Field -> (iii) \([MT^{-2}A^{-1}]\)\n(d) Permittivity of free space -> (iv) \([ML^{-1}T^{-2}]\)\n(v) \([MT^{-2}A^{-1}]\)
Pressure = \(F/A = [ML^{-1}T^{-2}]\); Angular Momentum = \(mvr = [ML^2T^{-1}]\); Magnetic field = \(F/qv = [MT^{-2}A^{-1}]\); Permittivity = \([M^{-1}L^{-3}T^4A^2]\).