A book with many printing errors contains four different expressions for the displacement 'y' of a particle executing simple harmonic motion. The wrong formula on dimensional basis
(i) `y=A sin (2pi t//T)`
(ii) `y=A sin (Vt)`
(iii) `y=A//T sin (t//A)`
(iv) `y=(A)/(sqrt(2))(sin omega t+cos omega t)`

Options

A. ii only
B. ii and iii only
C. iii only
D. iii and iv only

Answer

To determine which of the four expressions for the displacement 'y' of a particle executing simple harmonic motion are dimensionally incorrect, we will analyze each expression step by step.

### Step 1: Analyze the first expression
**Expression:** \( y = A \sin\left(\frac{2\pi t}{T}\right) \)

- Here, \( A \) is the amplitude, which has the dimension of length \([L]\).
- The term \( \frac{2\pi t}{T} \) is a dimensionless quantity because \( t \) (time) has the dimension \([T]\) and \( T \) (time period) also has the dimension \([T]\).
- Since the sine function is dimensionless, the entire expression \( y \) retains the dimension of \( A \), which is \([L]\).

**Conclusion:** This expression is dimensionally correct.

### Step 2: Analyze the second expression
**Expression:** \( y = A \sin(Vt) \)

- Here, \( A \) again has the dimension of length \([L]\).
- The term \( Vt \) must also be dimensionless for the sine function to be valid.
- The dimension of \( V \) (velocity) is \([L][T^{-1}]\) and \( t \) has the dimension \([T]\). Therefore, the dimension of \( Vt \) is:
\[ [L][T^{-1}] \cdot [T] = [L] \]
- Since \( Vt \) has the dimension of length, it is not dimensionless.

**Conclusion:** This expression is dimensionally incorrect.

### Step 3: Analyze the third expression
**Expression:** \( y = \frac{A}{T} \sin\left(\frac{t}{A}\right) \)

- Here, \( A \) has the dimension of length \([L]\) and \( T \) has the dimension of time \([T]\).
- The term \( \frac{t}{A} \) must be dimensionless. The dimension of \( t \) is \([T]\) and \( A \) is \([L]\), so:
\[ \frac{[T]}{[L]} = [T][L^{-1}] \]
- This is not dimensionless.

**Conclusion:** This expression is dimensionally incorrect.

### Step 4: Analyze the fourth expression
**Expression:** \( y = \frac{A}{\sqrt{2}} \left(\sin(\omega t) + \cos(\omega t)\right) \)

- Here, \( A \) has the dimension of length \([L]\).
- The term \( \omega t \) must also be dimensionless. The dimension of \( \omega \) (angular frequency) is \([T^{-1}]\) and \( t \) has the dimension \([T]\), so:
\[ \omega t = [T^{-1}][T] = 1 \quad (\text{dimensionless}) \]
- Since both sine and cosine functions are dimensionless, the entire expression retains the dimension of \( A \), which is \([L]\).

**Conclusion:** This expression is dimensionally correct.

### Final Conclusion
The expressions that are dimensionally incorrect are:
- Expression 2: \( y = A \sin(Vt) \)
- Expression 3: \( y = \frac{A}{T} \sin\left(\frac{t}{A}\right) \)

Thus, the answer is that the wrong formulas on a dimensional basis are expressions 2 and 3.

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