The dimensional formula of angular momentum is:
[1988]
Angular momentum is defined as \(L = r \times p\), where \(r\) is distance and \(p\) is linear momentum. Its dimensions are \([L][MLT^{-1}] = [ML^2T^{-1}]\).
The intervals measured by a clock given the following readings: 1.25 s, 1.24 s, 1.27 s, 1.21 s and 1.28 s. What is the percentage relative error in the observations?
[2020-Covid]
Mean value \(\bar{t} = (1.25+1.24+1.27+1.21+1.28)/5 = 1.25\text{ s}\). Mean absolute error \(\delta\bar{t} = (|0|+|0.01|+|0.02|+|0.04|+|0.03|)/5 = 0.1/5 = 0.02\text{ s}\). Percentage relative error \( = (\Delta\bar{t} / \bar{t}) \times 100% = (0.02 / 1.25) \times 100% = 1.6%\).
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 of (X), where \(X = \frac{A^2 B^{1/2}}{C^3 D^3}\) will be
[2019]
Assuming a common typo in the question and the formula should be \(X = \frac{A^2 B^{1/2}}{CD}): Percentage error in \(X = 2(\Delta A/A) + (1/2)(\Delta B/B) + (\Delta C/C) + (\Delta D/D)\). This gives \(2(1%) + (1/2)(2%) + 1(3%) + 1(4%) = 2% + 1% + 3% + 4% = 10%\).
In an experiment four quantities (a, b, c) and (d) are measured with percentage error 1%, 2%, 3% and 4% respectively. Quantity (P) is calculated as follows \(P = \frac{a^3 b^2}{cd}\). \(\text{%}\) error in (P) is:
[2013]
Percentage error in \(P = 3(\Delta a/a) + 2(\Delta b/b) + (\Delta c/c) + (\Delta d/d)\). Given errors: 3×1% + 2×2% + 1×3% + 1×4% = 3% + 4% + 3% + 4% = 14%.
A student measures the distance traversed in free fall of a body, initially at rest in a given time. He used this data to estimate (g), the acceleration due to gravity. If the maximum percentage errors in measurement of the distance and the time are \(e_1\) and \(e_2\) respectively, the percentage error in the estimation of (g) is:
[2010]
For free fall, \(s = \frac{1}{2}gt^2 \Rightarrow g = \frac{2s}{t^2}\). The percentage error in (g) is given by the sum of percentage errors of (s) and twice the percentage error of (t). So, Percentage error in \(g = e_1 + 2e_2\).
If the error in the measurement of radius of a sphere is 2%, then the error in the determination of volume of the sphere will be:
[2008]
Volume of a sphere \(V = \frac{4}{3}pi r^3\). The percentage error in volume is (3) times the percentage error in radius. Given \(\delta r/r \times 100 % = 2 % \). So, Percentage error in \(V = 3 \times 2 % = 6% \).
The error in measurement of radius of a sphere is 0.1% then error in its volume is:
Volume of a sphere \(V = \frac{4}{3}pi r^3\). The percentage error in volume is (3) times the percentage error in radius. Given \(\Delta r/r times 100% = 0.1%\). So, Percentage error in \(V = 3 \times 0.1% = 0.3% \).
The density of a cube is measured by measuring its mass and length of its sides. If the maximum error in the measurement of mass and lengths are 3% and 2% respectively, the maximum error in the measurement of density would be:
[1996]
Density of a cube \(\rho = M/L^3\). Percentage error in \( \rho = (\Delta M/M) + 3(\Delta L/L)\). Given \(\Delta M/M \times 100% = 3%\) and \(\Delta L/L \times 100% = 2%\). So, Percentage error in \(\rho \) = 3% + 3(2%) = 3% + 6% = 9% .
Percentage errors in the measurement of mass and speed are 2% and 3% respectively. The error in the estimate of kinetic energy obtained by measuring mass and speed will be:
[1995]
Kinetic energy \(K = \frac{1}{2}mv^2\). Percentage error in \(K = (\Delta m/m) + 2(\Delta v/v)\). Given \(Delta m/m \times\) 100% = 2% and \(\Delta v/v \times 100\)% = 3%. So, Percentage error in K = 2% + 2(3%) = 2% + 6% = 8%.
A certain body weighs 22.42 g and has a measured volume of 4.7 cc. The possible, error in the measurement of mass and volume are 0.01 g and 0.1 cc. Then maximum error in the density will be:
[1991]
Density \(\rho = M/V\). Fractional error \(\Delta\rho/rho = (\Delta M/M) + (\Delta V/V)\). Given \(M=22.42\text{ g}\), \(\Delta M=0.01\text{ g}\). \(V=4.7\text{ cc}\), \(\Delta V=0.1\text{ cc}\). So, \(Delta M/M = 0.01/22.42 \approx 0.000446\). \(\Delta V/V = 0.1/4.7 \approx 0.02127\). Total fractional error \(\approx 0.021716\). Percentage error \(\approx 2.17%\), which is closest to 2%.