In equation, `r=m^(2)sin pit`, where t represent time if the unit of m is N, then the unit of r is

Options

A. N
B. `N^(2)`
C. `Ns`
D. `N^(2)s`

Explanation

To solve the problem step by step, we need to analyze the given equation and the units involved.

### Step 1: Understand the equation
The equation given is \( r = m^2 \sin(\pi t) \). Here, \( r \) is the variable we want to find the unit for, \( m \) is given in Newtons (N), and \( t \) represents time.

### Step 2: Identify the unit of \( m \)
We know that \( m \) is given in Newtons (N). The unit of force (Newton) can be expressed in terms of base SI units:
\[ 1 \, \text{N} = 1 \, \text{kg} \cdot \text{m/s}^2 \]
Thus, the unit of \( m \) is \( \text{N} \).

### Step 3: Determine the unit of \( m^2 \)
Next, we need to find the unit of \( m^2 \):
\[ m^2 = (\text{N})^2 = (1 \, \text{kg} \cdot \text{m/s}^2)^2 = 1 \, \text{kg}^2 \cdot \text{m}^2/\text{s}^4 \]
So, the unit of \( m^2 \) is \( \text{kg}^2 \cdot \text{m}^2/\text{s}^4 \).

### Step 4: Analyze the sine function
The sine function, \( \sin(\pi t) \), is a dimensionless quantity. This means it does not contribute any units to the equation.

### Step 5: Combine the units
Since \( r = m^2 \sin(\pi t) \) and \( \sin(\pi t) \) is dimensionless, the unit of \( r \) is simply the unit of \( m^2 \):
\[ \text{Unit of } r = \text{Unit of } m^2 = \text{kg}^2 \cdot \text{m}^2/\text{s}^4 \]

### Conclusion
Thus, the unit of \( r \) is \( \text{kg}^2 \cdot \text{m}^2/\text{s}^4 \).

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