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.
Assertion (A): Light year and wavelength have same dimensions.
Reason (R): Light year represent time while wavelength represent distance.
Both light year and wavelength represent distance and have the dimension \([L]\). Since light year represents distance and not time, R is false.
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.
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 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. \([\text{Energy density}] = [\text{Energy}]/[\text{Volume}] = (ML^2T^{-2})/L^3 = ML^{-1}T^{-2}\).
Which two of the following five physical parameters have the same dimensions?
A. Energy density
B. Refractive index
C. Dielectric constant
D. Young’s modulus
E. Magnetic field
[2008]
Energy density (A) is \(ML^{-1}T^{-2}\). Young's modulus (D) is Stress/Strain, so also \(ML^{-1}T^{-2}\). Therefore, (A) and (D) have the same dimensions.
Dimensions of resistance in an electrical circuit, in terms of dimension of mass \(M\), of length \(L\), of time \(T\) and of current \(I\), would be:
[2007]
Resistance \(R = V/I = (W/q)/I = W/(I^2T)\). Work \([W] = ML^2T^{-2}\). So, \([R] = (ML^2T^{-2})/(I^2T) = ML^2T^{-3}I^{-2}\).