Taking 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 \(9.98\text{ m}\).
The pitch of a screw gauge is 1 mm and there are 100 divisions on circular scale. While measuring the thickness of a sheet, the main scale reads 1 mm and \(52^{\text{nd}}\) division on circular scale coincide with the reference line. The thickness of the sheet is
Least count \(LC = \frac{\text{Pitch}}{\text{Number of circular divisions}} = \frac{1 \text{ mm}}{100} = 0.01 \text{ mm}\). Thickness \(= \text{MSR} + \text{CSR} \times LC = 1 \text{ mm} + 52 \times 0.01 \text{ mm} = 1.52 \text{ mm} = 0.152 \text{ cm}\).
Dashrath measures the length of a wire using a meter scale with a least count of 1 mm and finds it to be L = 75.0 cm. He also measures diameter of thin wire using a screw gauge with a least count of 0.01 mm and finds it to be d = 0.500 cm. He uses these measurements to calculate the volume of wire. The maximum percentage error in volume of wire is nearly
Volume of wire is \(V = \pi \left(\frac{d}{2}\right)^2 L ⇒ \frac{\Delta V}{V} = 2\frac{\Delta d}{d} + \frac{\Delta L}{L} = 2\left(\frac{0.001}{0.500}\right) + \frac{0.1}{75.0} \approx 0.53%\).
The dimensional formula for impulse is
Impulse is given by \(I = F \cdot t = [MLT^{-2}][T] = [MLT^{-1}]\).
Assertion (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.
Assertion (A): When an algebraic equation has been derived, it is advisable to check it for dimensional consistency.
Reason (R): This guarantees that the equation is correct.
A dimensionally consistent equation is not guaranteed to be physically correct, as dimensionless constants cannot be verified through dimensional analysis. Thus, R is false.
Assertion (A): eV and joule are the S.I. units of energy used in modern physics and mechanics respectively.
Reason (R): Different types of energies require different units in S.I. units.
The SI unit of all forms of energy is the joule. Electron-volt (eV) is not an SI unit, and different forms of energy do not require different SI units.
Assertion (A): Pressure and energy density have same units in SI.
Reason (R): Dimensions of energy density and pressure are same.
Pressure has dimensions of \([M L^{-1} T^{-2}]\). Energy density (energy per unit volume) also simplifies to \([M L^{-1} T^{-2}]\). Since their dimensions are identical, they share the same SI units.
Assertion (A): The dimensions of base (fundamental) quantity in other base quantities is always zero.
Reason (R): All derived quantities may be represented dimensionally in terms of fundamental quantities.
Base quantities are mutually independent and cannot be defined in terms of each other, making the exponent of one base quantity in another zero. Both statements are true, but R is not the explanation of A.
Assertion (A): A unitless quantity never has a non-zero dimension.
Reason (R): A dimensionless quantity never has a unit.
A quantity without a unit is always dimensionless, so A is true. However, a dimensionless quantity can have a unit (for example, plane angle has the unit radian), making R false.