Question

Given that$y=a\cos \left( \dfrac{t}{p}-qx \right)$ , where t represents time in second and x represents distance in meters. Which of the following statements is true?
a) The unit of x is same as that of q
b) The unit of x is same as that of p
c) The unit of t is same as that of q
d) The unit of t is same as that of p

Answer 1

Whenever two quantities are given as equal to one another think in terms of dimensions. The quantities will be the same if and only if their dimensions are the same. So, to solve these types of questions make sure that both the quantities on the left side as well as right side are equal. In this case they are dimensionless.

Complete step by step answer:
We have an expression as: $y=a\cos \left( \dfrac{t}{p}-qx \right)$
Since $\left( \dfrac{t}{p}-qx \right)$ is a dimensionless quantity. Therefore, $\dfrac{t}{p}$ and $qx$ are also dimensionless.
As t represents time, so for the quantity $\dfrac{t}{p}$ to be dimensionless, p has the same dimensions as t. Similarly, for quantity $qx$ to be dimensionless, q has the same dimensions as x.
Therefore, option (a) and option (d) are the correct answer.
Additional information:
Dimensions of a physical quantity are the powers to which the fundamental units are raised to obtain one unit of that quantity.Dimensional analysis is the practice of checking relations between physical quantities by identifying the dimensions of the physical quantities.

Note: Dimensions help us in making sure that the quantiles are equal to one another. It works like a weighing balance helping us find out the type of quantities that are equated to one another.