Step 1: Understanding the Concept:
The principle of homogeneity of dimensions states that the dimensions of each term in a physical equation must be the same.
Furthermore, the arguments of trigonometric functions (like sine, cosine) must be dimensionless because they represent angles.
: Key Formula or Approach:
1. The displacement \( y \) has the dimensions of length \( [L] \).
2. The argument \( [Bx + Ct + D] \) (as corrected in the solution key) must be dimensionless \( [M^0 L^0 T^0] \).
3. Every individual term within the bracket must be dimensionless.
Step 2: Detailed Explanation:
From the given equation \( y = A \sin(Bx + Ct + D) \):
- Since \( \sin(\theta) \) is a dimensionless number, the dimensions of \( A \) must match \( y \).
\[ [A] = [y] = [L] \]
- The term \( Bx \) must be dimensionless. Since \( x \) is position \( [L] \):
\[ [B][L] = [1] \implies [B] = [L^{-1}] \]
- The term \( Ct \) must be dimensionless. Since \( t \) is time \( [T] \):
\[ [C][T] = [1] \implies [C] = [T^{-1}] \]
- The term \( D \) is a constant added to dimensionless quantities, so \( D \) itself is dimensionless.
\[ [D] = [1] \]
Now, calculate the dimensions of the product \( [ABCD] \):
\[ [ABCD] = [A][B][C][D] \]
\[ [ABCD] = [L] \cdot [L^{-1}] \cdot [T^{-1}] \cdot [1] \]
\[ [ABCD] = [L^0 T^{-1}] \]
Expressing in standard \( M, L, T \) form:
\[ [ABCD] = [M^0 L^0 T^{-1}] \]
Step 3: Final Answer:
The dimensional formula for \( [ABCD] \) is \( [M^0 L^0 T^{-1}] \).