The equation of a S.H.M. of amplitude `A` and angular frequency `omega` in which all distances are measured from one extreme position and time is taken to be zero at the other extreme position is

Options

A. `x = A sin omega t`
B. `x = A (cos omega t + sin omega t)`
C. `x = A-A cos omega t`
D. `x = A + A cos omega t`

Answer

To derive the equation of Simple Harmonic Motion (S.H.M.) given the conditions in the question, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Basics of S.H.M.:**
The general equations for S.H.M. can be expressed as:
\[ x(t) = A \cos(\omega t + \phi) \]
or
\[ x(t) = A \sin(\omega t + \phi) \]
where \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( \phi \) is the phase constant.

2. **Setting the Reference Frame:**
In this problem, we are measuring all distances from one extreme position. Let's say the extreme position we are measuring from is \( x = 0 \) (the maximum displacement in one direction). The other extreme position will then be \( x = 2A \) (the maximum displacement in the opposite direction).

3. **Choosing the Initial Conditions:**
We are told that time \( t = 0 \) corresponds to the other extreme position (which is \( x = 2A \)). Therefore, we need to modify our equation to reflect this condition.

4. **Formulating the Equation:**
Since we are measuring from the extreme position at \( x = 0 \), we can express the position \( x \) as:
\[ x(t) = 2A - A \cos(\omega t) \]
This equation indicates that at \( t = 0 \), \( x(0) = 2A - A \cos(0) = 2A - A = A \), which is indeed the initial position we are measuring from.

5. **Finalizing the Equation:**
To express the position in terms of the amplitude and angular frequency, we can rewrite the equation as:
\[ x(t) = A + A \cos(\omega t) \]
or equivalently,
\[ x(t) = A(1 + \cos(\omega t)) \]

### Conclusion:
The equation of S.H.M. of amplitude \( A \) and angular frequency \( \omega \), where all distances are measured from one extreme position and time is taken to be zero at the other extreme position, is:
\[ x(t) = A + A \cos(\omega t) \]