Question 1:
easyThe 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.
Question 2:
easyIn 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: \(\frac{1}{3}(0.3) + 2(0.2) + \frac{1}{2}(0.6) = 0.1 + 0.4 + 0.3 = 0.8\).
Question 3:
easyWeight 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 \(\frac{2+0+2+3+3}{5} = 2\text{ N}\).
Question 4:
easyThe side of a cube is \( 2.00 + 0.01\text{ cm}\). The volume and total surface area of cube respectively are
Volume \(V = a^3 = 8.00\text{ cm}^3\), \(\Delta V = 3 V \frac{\Delta a}{a} = 0.12\text{ cm}^3\). Surface Area \(S = 6a^2 = 24.0\text{ cm}^2\), \(\Delta S = 2 S \frac{\Delta a}{a} = 0.24\text{ cm}^2\).
Question 5:
easyGiven below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): The absolute error has the same unit as the quantity itself.
Reason (R): Fractional error has no unit.
In the light of above statements, choose the correct answer from the options given below.
Absolute error \(\Delta x\) has the same unit as the physical quantity. Fractional error is the ratio \(\Delta x / x\) and is dimensionless (no unit). Both statements are true, but the lack of unit in fractional error does not explain why absolute error has a unit.
Question 6:
easyTaking into account of the significant figures, value of \(9.99\text{ m} – 0.0099\text{ m}\) is
When subtracting, the result must be rounded off to the least number of decimal places in any of the terms. Here, \(9.99\) has two decimal places, so \(9.9801\) is rounded to \(9.98\text{ m}\).
Question 7:
easyAssertion (A): The error in measurement of radius of the sphere is 0.3%. The permissible error in its surface area is 1.2%.
Reason (R): Area of sphere, \(A = 4\pi r^2 \Rightarrow \frac{\Delta A}{A} = 4\frac{\Delta r}{r}\).
The surface area of a sphere is \(A = 4\pi r^2\), which gives the fractional error relation as \(\frac{\Delta A}{A} = 2\frac{\Delta r}{r}\). Thus, the error in area is \(2 \times 0.3% = 0.6%\), making both statements false.
Question 8:
easyAssertion (A): Mean absolute error of a measurement is always positive.
Reason (R): Mean absolute error is defined as the magnitude of difference between true value and measured value.
Mean absolute error is the arithmetic mean of all absolute errors and is always positive. The definition given in the reason describes individual absolute error rather than the mean absolute error, so R is false.
Question 9:
easyAssertion (A): If the measuring instruments used are perfect, then measurements made will be perfect.
Reason (R): Measurements depend upon only on the instruments.
Even with perfect instruments, errors due to observation, environmental factors, or experimental setup can occur. Both statements are false.
Question 10:
easyTwo quantities are measured as \( P = (1 \pm 0.40) \, \text{m} \), \( Q = (4 \pm 0.20) \, \text{m} \). The correct value of \( (PQ)^{1/2} \) will be
Let \( Y = (PQ)^{1/2} \). Its value is \( Y = (1 \times 4)^{1/2} = 2 \, \text{m} \). The relative error is \( \frac{\Delta Y}{Y} = \frac{1}{2} \left(\frac{\Delta P}{P} + \frac{\Delta Q}{Q}\right) = \frac{1}{2}\left(\frac{0.40}{1} + \frac{0.20}{4}\right) = 0.225 \), giving \( \Delta Y = 0.45 \, \text{m} \). Thus, \( Y = (2 \pm 0.45) \, \text{m} \).