Given that: \(y=A \sin \left[\left(\frac{2 \pi}{\lambda}\right)(c t-x)\right]\), where \(y\) and \(x\) are…
Given that: \(y=A \sin \left[\left(\frac{2 \pi}{\lambda}\right)(c t-x)\right]\), where \(y\) and \(x\) are measured in the unit of length, Which of the following statements is true?
The unit of \(\lambda\) is same as that of \(x\) and \(A\).
The unit of \(\lambda\) is same as that of \(x\) but not of \(A\).
The unit of \(c\) is same as that of \(2 \pi / \lambda\)
The unit of \((c t-x)\) is same as that of \(2 \pi / \lambda\)
Solution
Here, unit of \(y\) and \(A\) will be same and that of \(x\) and \(\lambda\) will be same. \(\frac{2 \pi}{\lambda}(c t-x)\) is dimensionless. Here, \(\frac{c t}{\lambda}\) and \(\frac{x}{\lambda}\) are dimensionless. Unit of \(c t\) is same as that of \(\lambda\) or \(x .\)
Unit of ct must be same as unit of \(x\). Since argument of sin has to be dimensionless, thus dimensions of \(x\) and ct must be same as dimensions of \(\lambda\). Thus dimension of \(\lambda\) equals ' \(L\) '. Also, dimension of \(y\) equals ' \(L\) ' and hence dimension of \(A\) equals ' \(L\) ' because \(\sin\) term is dimensionless. Thus statement 1 is correct.
Asked in: JEE Mains - Units and Dimensions - Chapter Test