A student writes four different expressions for the displacement `y` in a periodic motion . Which of the following can be correct?

Options

A. ` y = a T sin ( 2 pi t)/(T)`
B. ` y = a sin V t`
C. ` y = (a)/( T) sin ( t)/(a)`
D. ` y = (a)/ (sqrt(2))[ sin (2pi t)/(T) + cos (2pi t)/(T)]`

Explanation

To determine which expressions for displacement \( y \) in periodic motion can be correct, we need to analyze each expression based on dimensional analysis. The key point is that any angle used in trigonometric functions must be dimensionless.

### Step-by-Step Solution:

1. **Understanding the Displacement in Periodic Motion**:
Displacement in periodic motion can often be expressed in terms of sine or cosine functions. The general form is:
\[ y = A \sin(\omega t + \phi) \]
where \( A \) is the amplitude, \( \omega \) is the angular frequency, \( t \) is time, and \( \phi \) is the phase constant.

2. **Analyzing the First Expression**:
Consider the expression:
\[ y = A t \sin\left(\frac{2\pi t}{T}\right) \]
- Here, \( \frac{2\pi t}{T} \) is the argument of the sine function.
- Dimensions of \( t \) are [T] (time) and \( T \) is also [T].
- Therefore, \( \frac{t}{T} \) is dimensionless, but \( A t \) has dimensions of [L][T].
- Since the sine function requires a dimensionless argument, this expression is **not correct**.

3. **Analyzing the Second Expression**:
Consider:
\[ y = A \sin(vt) \]
- Here, \( v \) is velocity with dimensions [L][T]⁻¹.
- Thus, \( vt \) has dimensions of [L], which is not dimensionless.
- Therefore, this expression is also **not correct**.

4. **Analyzing the Third Expression**:
Consider:
\[ y = A \sin(t) \]
- Here, \( t \) has dimensions of [T].
- Since \( t \) is not dimensionless, this expression is **not correct**.

5. **Analyzing the Fourth Expression**:
Consider:
\[ y = \frac{A}{\sqrt{2}} \sin\left(\frac{2\pi t}{T}\right) + \cos\left(\frac{2\pi t}{T}\right) \]
- The arguments \( \frac{2\pi t}{T} \) are dimensionless since \( t \) and \( T \) both have dimensions of time.
- Therefore, both sine and cosine functions have dimensionless arguments, making this expression **correct**.

### Conclusion:
The only correct expression for displacement \( y \) in periodic motion from the given options is:
\[ y = \frac{A}{\sqrt{2}} \sin\left(\frac{2\pi t}{T}\right) + \cos\left(\frac{2\pi t}{T}\right) \]