Assertion (A): Only like quantities can be added or subtracted from each other.
Reason (R): Velocity can be subtracted from the velocity gradient.
By the principle of homogeneity, only physical quantities with identical dimensions can be added or subtracted. Velocity and velocity gradient have different dimensions, hence they cannot be subtracted.
Assertion (A): If a physical quantity has a unit it must have dimension.
Reason (R): There may exist a physical quantity which has dimension but no unit.
An angle has a unit (radian) but is dimensionless, so A is false. Any physical quantity that possesses dimensions must have a unit, so R is also false.
The vernier scale of a callipers is divided into 20 divisions which coincide with 18 main scale divisions. Each main scale division is 0.2 mm. The least count of the instrument is
Least count (LC) is given by \(text{LC} = 1text{ MSD} - 1text{ VSD}\). Since 20 VSD = 18 MSD, we have \(1text{ VSD} = 0.9text{ MSD}\). Thus, \(text{LC} = 0.1text{ MSD} = 0.1 times 0.2text{ mm} = 0.02text{ mm}\).
Two 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, \( Y = (2 \pm 0.45) \, \text{m} \).
Dashrath measures the length of a wire using a meter scale with a least count of \( 1\text{ mm} \) and finds it to be \( L = 75.0\text{ cm} \). He also measures diameter of thin wire using a screw gauge with a least count of \( 0.01\text{ mm} \) and finds it to be \( d = 0.500\text{ cm} \). He uses these measurements to calculate the volume of wire. The maximum percentage error in volume of wire is nearly
Volume \( V = \frac{\pi d^2 L}{4} ⇒ \frac{\Delta V}{V} = 2 \frac{\Delta d}{d} + \frac{\Delta L}{L} \). Here, \( \Delta d = 0.001\text{ cm} \) and \( \Delta L = 0.1\text{ cm} \). Thus, \( \frac{\Delta V}{V} = 2\left(\frac{0.001}{0.500}\right) + \frac{0.1}{75} = 0.4% + 0.13% = 0.53% \).
The dimensional formula for impulse is
Impulse is defined as Force multiplied by time, \( I = F \cdot t \). Its dimensions are \( [\text{MLT}^{-2}][\text{T}] = [\text{MLT}^{-1}] \).
Assertion (A): In SHM let \(x\) be the maximum speed, \(y\) the frequency of oscillation and \(z\) the maximum acceleration, then \(\frac{xy}{z}\) is a constant quantity.
Reason (R): This is because \(\frac{xy}{z}\) becomes a dimensionless quantity
For SHM, \(x=A\omega\), \(y=\frac{\omega}{2\pi}\), \(z=A\omega^2\). Thus, \(\frac{xy}{z} = \frac{(A\omega)(\omega/(2\pi))}{(A\omega^2)} = \frac{1}{2\pi}\), which is a constant. So (A) is true. The dimensions are \([x]=LT^{-1}\), \([y]=T^{-1}\), \([z]=LT^{-2}\), making \([xy/z]=1\), dimensionless. So (R) is true. However, being dimensionless does not explain why it's a constant.
The angle \(1’\) (minute of arc) in radian is nearly equal to,
[2020- Covid]
Convert \(1'\) to degrees, then to radians. \(1' = (1/60)^\circ = (1/60) \times (\pi/180)\text{ rad} \approx 2.91 \times 10^{-4}\text{ rad}\)
The unit of thermal conductivity is
[2019]
Thermal conductivity \(k\) relates heat flux to temperature gradient. So, \(k\) has units of \(W/(m \cdot K) = W m^{-1} K^{-1}\).
The damping force on an oscillator is directly proportional to the velocity. The units of the constant of proportionality are:
[2012 pre]
Damping force \(F_d = -bv\). The constant \(b\) has dimensions \([F_d]/[v] = (MLT^{-2})/(LT^{-1}) = MT^{-1}\), which is \(kg s^{-1}\).