Assertion (A): Angle and strain are dimensionless.
Reason (R): Angle and strain have no unit.
Both angle and strain are ratios of like quantities and are dimensionless. However, angle has a unit (radian), so R is false.
Assertion (A): A displacement can be added with a distance.
Reason (R): Adding a scalar to a vector of the same dimensions is a meaningful algebraic operation.
Displacement is a vector quantity while distance is a scalar quantity. Scalars and vectors cannot be added directly even if they have the same dimensions, making both statements false.
Assertion (A): If \(vec{r}\) is the position vector then dimensions of \(\frac{d^2vec{r}}{dt^2}\) is \([M^0L^1T^{-2}]\).
Reason (R): Dimensions of \(int \left(\frac{d^2vec{r}}{dt^2}\right) dt\) is \([M^0L^1T^{-1}]\) where \(\vec{r} \rightarrow\) position vector, \(t \rightarrow\) time.
The second derivative of position with respect to time represents acceleration with dimensions \([L T^{-2}]\). Integrating this acceleration with respect to time yields velocity with dimensions \([L T^{-1}]\). Both statements are true, but R is not the explanation of A.
Assertion (A): In mechanics the method of dimensions can’t be applied to derive formula of a physical quantity which depends on more than three physical quantities.
Reason (R): We can derive relation of a physical quantity with other physical quantities out of which two have same dimensions.
In mechanics, we have only three base dimensions (M, L, T). Thus, we cannot determine more than three independent exponents. If two quantities have the same dimensions, they cannot be resolved independently, making R false.
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.
Dimensions of stress are:
[2020]
Stress is Force per unit Area. \([\text{Stress}] = [F]/[A] = (MLT^{-2})/L^2 = ML^{-1}T^{-2}\).
The dimensions of \((\mu_0\epsilon_0)^{-1/2}\) are:
[2012 Mains]
The speed of light \(c = 1/\sqrt{\mu_0\epsilon_0}\). Therefore, \((\mu_0\epsilon_0)^{-1/2} = c\). The dimensions of speed are \([LT^{-1}]\).
The dimension of \(\frac{1}{2}\epsilon_0 E^2\), where \(\epsilon_0\) is permittivity of free space and \(E\) is electric field, is:
[2010 Pre]
The expression \(\frac{1}{2}\epsilon_0 E^2\) represents electric energy density, which is Energy per unit Volume. \([\text{Energy density}] = [\text{Energy}]/[\text{Volume}] = (ML^2T^{-2})/L^3 = ML^{-1}T^{-2}\).
The dimensions of universal gravitational constant are:
[2004]
From \(F = G \frac{m_1 m_2}{r^2}\), we get \(G = \frac{Fr^2}{m_1 m_2}\). So, \([G] = \frac{(MLT^{-2})(L^2)}{M^2} = M^{-1}L^3T^{-2}\).