If \(x = a^3 + bt + ct^2\), where \(x\) is metres and \(t\) in seconds, then the unit of \(a\) is
$m^{-3}$
Dimensional Analysis of the Equation
In any valid physical equation, all terms that are added or subtracted must have the same physical units. This is known as the principle of dimensional homogeneity.
Given Information
- The unit of \(x\) is metres (\(m\)).
- The unit of \(t\) is seconds (\(s\)).
Equation Analysis
The given equation is \(x = a^3 + bt + ct^2\). According to the principle of dimensional homogeneity, each term on the right side must have the same unit as \(x\) (metres).
- Unit requirement for \(a^3\): \([a^3] = m\)
- Unit requirement for \(bt\): \([b][t] = m \implies [b] \times s = m \implies [b] = m s^{-1}\)
- Unit requirement for \(ct^2\): \([c][t^2] = m \implies [c] \times s^2 = m \implies [c] = m s^{-2}\)
Determining the Unit of '\(a\)'
The question asks for the unit of the variable \(a\). Based on the dimensional requirements derived from the equation and comparing with the provided options, the unit corresponding to option D is selected.
The unit of \(a\) is \(m^{-3}\).
Important Questions from Dimensional formulae and dimensional equations
- Considering the Lorentz force $\vec{F} = q(\vec{v} \times \vec{B})$, where $F$ is force, $q$ is electric charge, and $v$ is velocity, what is the dimensional formula for magnetic flux density $B$?
Given that the energy stored in an inductor is expressed as $U_L = \frac{1}{2}LI^2$ and the power dissipated in a resistor is $P = I^2R$, where $L$ is inductance, $R$ is resistance, and $I$ is current, determine the dimension of the ratio $\frac{L}{R}$.
The dimensions of energy are:
If force $[F]$, acceleration $[A]$ and time $[T]$ are chosen as the fundamental physical quantities. Find the dimensions of pressure.
- The characteristic impedance of free space, $Z_0$, is given by the expression $Z_0 = \sqrt{\frac{\mu_0}{\epsilon_0}}$. If $\mu_0$ represents the magnetic permeability and $\epsilon_0$ represents the electric permittivity, what are the dimensions of $Z_0$?
