Sheet 05 Errors In Measurement and Combination of Errors.
The radius of a thin metallic wire measured with screw gauge, the values of the radius of the wire are 3.04mm, 3.06mm, 3.08mm, 3.07mm and 3.05mm.Find: (i) Average radius of the wire? (ii) Absolute error in each observation? (iii) Mean absolute error? (iv) Relative error? and (v) Percentage Error?
Ans:(i) \(\bar{R}=3.06mm\) (ii)0.02 mm, 0.00 mm, -0.02 mm, -0.01 mm and 0.01 mm.(iii) \(\Delta\bar{R}=0.012mm\) (iv) \(\frac{\Delta\bar{R}}{\bar{R}}=0.0039\) (v) \(\frac{\Delta\bar{R}}{\bar{R}}\times100\%=0.39\%\).
The refractive index of water in an experiment was observed to be 1.30, 1.34, 1.35, 1.32, 1.36 and 1.33. Calculate: (i) Mean value of refractive index? (ii) Mean absolute error? (iii) Fractional/relative error? (iv) Percentage Error? (v) Express the result in terms of absolute error and percentage error.
Ans:(i) \(\bar{\mu}=1.33\) (ii) \(\Delta\bar{\mu}=0.016\) (iii) \(\frac{\Delta\bar{\mu}}{\bar{\mu}}=0.012\) (iv) \(\frac{\Delta\bar{\mu}}{\bar{\mu}}\times100\%=1.20\%\) (iv) \(\left(1.33\pm0.016\right)\) and \(\left(1.33\pm1.20\%\right)\).
The time period of oscillation of a simple pendulum in an experiment is recorded as 2.56s, 2.62s, 2.70s, 2.58s, 2.45s respectively. Find: (i)Time period. (ii) Mean absolute error. (iii) Percentage Error.
Ans:(i) \(\bar{T}=2.58s\) (ii) \(\Delta\bar{T}=0.062s\) and (iii) \(\frac{\Delta\bar{T}}{\bar{T}}\times100\%=2.4\%\).
In an experiment the focal length of a concave mirror found to have values in successive measurement as 17.3cm, 17.7cm, 18.4cm, 18.3cm, 17.9am and 18.0cm. Calculate: (i) The mean absolute error? and (ii) Percentage error? (iii) Also express the final value proper way.
Ans:(i) \(\Delta\bar{f}=0.25cm\) (ii) \(\frac{\Delta\bar{f}}{\bar{f}}\times100\%=1.4\%\) and (iii) \(f=\left(17.9\pm0.25\right)cm\) and \(f=\left(17.9\pm1.4\%\right)cm\).
The temperature of two bodies measured by a thermometer are \(t_{1}=20^{o}C\pm 0.5^{o}C\) and \(t_{2}=50^{o}C\pm 0.5^{o}C\). What will be the temperature difference and error there in?
Ans: \(\Delta t=30^{o}C\pm1^{o}C\).
The length and breadth of a rectangular area are \(\left(11.4\pm 0.2\right)cm\) and \(\left(6.8\pm 0.4\right)cm\). Calculate: the area of rectangular area with error limits.
Ans: \(A=\left(77.52\pm4.6\right)cm^{2}\).
If, the length and time period of an oscillating pendulum have errors \(3\%\) and \(2\%\) respectively, Then what is the error in estimate of g?
Ans: \(\frac{\Delta g}{g}\%=7\%\).
The time period of oscillations of a simple pendulum is \(T=2\pi\sqrt{\frac{L}{g}}\). Measured value of L is \(20.0cm\) known to \(1mm\) accuracy and time for 100 oscillations of the pendulum is found to be \(90s\) using a wrist watch of \(1s\) resolution. What is accuracy in determining g?
Ans: \(\frac{\Delta g}{g}\times100\%=2.72\%\).
The mass and density of a solid sphere are measured to be \(\left(12.4\pm 0.1\right)kg\) and \(\left(4.6\pm 0.2\right) kg.m^{-3}\). Calculate: The volume of the sphere with error limits.
