Which of the following is/are not SHM?

Options

A. `y= A cos omega t`
B. `y = A sin omega t`
C. `y =A sin 3 omega t`
D. `y= A e^(kT)`

Answer

To determine which of the given options is not Simple Harmonic Motion (SHM), we need to analyze the equations provided and check if they satisfy the condition for SHM. The condition for SHM is given by the equation:

\[ \frac{d^2x}{dt^2} = -\omega^2 x \]

This means that the acceleration of the object is directly proportional to its displacement from the equilibrium position and is always directed towards that position.

### Step 1: Analyze the first option
Let's consider the first option, which is \( x = A \cos(\omega t) \).

1. Differentiate \( x \) with respect to time \( t \):
\[ \frac{dx}{dt} = -A \omega \sin(\omega t) \]

2. Differentiate again to find acceleration:
\[ \frac{d^2x}{dt^2} = -A \omega^2 \cos(\omega t) \]

3. Substitute \( x = A \cos(\omega t) \) into the SHM equation:
\[ \frac{d^2x}{dt^2} = -\omega^2 x \]

This shows that this option satisfies the SHM condition.

### Step 2: Analyze the second option
Now, let's consider the second option, which is \( x = A \sin(\omega t) \).

1. Differentiate \( x \) with respect to time \( t \):
\[ \frac{dx}{dt} = A \omega \cos(\omega t) \]

2. Differentiate again to find acceleration:
\[ \frac{d^2x}{dt^2} = -A \omega^2 \sin(\omega t) \]

3. Substitute \( x = A \sin(\omega t) \) into the SHM equation:
\[ \frac{d^2x}{dt^2} = -\omega^2 x \]

This option also satisfies the SHM condition.

### Step 3: Analyze the third option
Next, consider the third option, which is \( x = A e^{kt} \).

1. Differentiate \( x \) with respect to time \( t \):
\[ \frac{dx}{dt} = A k e^{kt} \]

2. Differentiate again to find acceleration:
\[ \frac{d^2x}{dt^2} = A k^2 e^{kt} \]

3. Substitute \( x = A e^{kt} \) into the SHM equation:
\[ \frac{d^2x}{dt^2} = -\omega^2 x \]

This gives:
\[ A k^2 e^{kt} = -\omega^2 A e^{kt} \]

If we cancel \( A e^{kt} \) (assuming \( A \neq 0 \)), we get:
\[ k^2 = -\omega^2 \]

This is not possible since both \( k^2 \) and \( \omega^2 \) are positive quantities. Therefore, this option does not satisfy the SHM condition.

### Conclusion
Thus, the option that is not SHM is:

**D) \( x = A e^{kt} \)**