In equation `y=x^(2)cos^(2)2pi(betagamma)/alpha`, the units of `x,alpha,beta` are m, `s^(-1)and(ms^(-1))^(-1)` respectively. The units of y and `gamma` are

Options

A. `m^(2),ms^(-2)`
B. `m,ms^(-1)`
C. `m^(2),m`
D. `m,ms^(-2)`

Answer

To solve the problem, we need to analyze the given equation and the units of the variables involved. The equation provided is:

\[ y = x^2 \cos^2(2\pi \beta \gamma) / \alpha \]

We are given the units of \( x \), \( \alpha \), and \( \beta \) as follows:
- \( x \): meters (m)
- \( \alpha \): \( s^{-1} \) (per second)
- \( \beta \): \( (ms^{-1})^{-1} \) (reciprocal of meters per second)

### Step 1: Determine the units of \( \gamma \)

From the equation, we see that the argument of the cosine function, \( 2\pi \beta \gamma \), must be dimensionless because cosine is a function that operates on angles, which are dimensionless quantities (measured in radians). Therefore, the product \( \beta \gamma \) must also be dimensionless.

Given that:
- The unit of \( \beta \) is \( (ms^{-1})^{-1} = s/m \)

We can express \( \gamma \) in terms of \( \alpha \) and \( \beta \):

\[ \gamma = \frac{\text{dimensionless}}{\beta} = \frac{\alpha}{\beta} \]

Now substituting the units:
- The unit of \( \alpha \) is \( s^{-1} \)
- The unit of \( \beta \) is \( s/m \)

Thus, the unit of \( \gamma \) can be calculated as:

\[ \text{Unit of } \gamma = \frac{s^{-1}}{s/m} = \frac{m}{s^2} \]

So, the unit of \( \gamma \) is \( m \cdot s^{-2} \).

### Step 2: Determine the units of \( y \)

Now we need to find the units of \( y \). From the equation, we have:

\[ y = x^2 \cos^2(2\pi \beta \gamma) / \alpha \]

Since \( \cos^2(2\pi \beta \gamma) \) is dimensionless, we can ignore its contribution to the units. Thus, we can focus on the remaining part:

\[ y = \frac{x^2}{\alpha} \]

Substituting the units:
- The unit of \( x^2 \) is \( (m)^2 = m^2 \)
- The unit of \( \alpha \) is \( s^{-1} \)

Now we can calculate the unit of \( y \):

\[ \text{Unit of } y = \frac{m^2}{s^{-1}} = m^2 \cdot s \]

Thus, the unit of \( y \) is \( m^2 \cdot s \).

### Final Answer
- The unit of \( y \) is \( m^2 \cdot s \).
- The unit of \( \gamma \) is \( m \cdot s^{-2} \).