Question

The displacement of the particle at a distance \(x\) from the origin is given by \(Y = A \sin \omega \left(\dfrac{x}{v} - k\right)\), where \(\omega\) is the angular velocity and \(v\) is the linear velocity. The dimensions of \(k\) are

• A. \([L^0 M^0 T^1]\)
• B. \([L^1 M^0 T^1]\)
• C. \([L^0 M^0 T^{-1}]\)
• D. \([L^0 M^0 T^2]\)

Hint:

Always ensure arguments of sine and cosine are dimensionless.

Answer 1

The Correct Option is A

Step 1: Argument of sine function.
The argument of a trigonometric function must be dimensionless.
Step 2: Dimensions of \(\dfrac{x}{v}\).
\[ [x] = L, \quad [v] = LT^{-1} \Rightarrow \left[\frac{x}{v}\right] = T \]
Step 3: Role of constant \(k\).
Since \(\left(\dfrac{x}{v} - k\right)\) must have same dimensions,
\[ [k] = T \]
Step 4: Conclusion.
\[ [k] = [L^0 M^0 T^1] \]