A particle moves on the X-axis according to the equation `x=A+Bsinomegat`. Let motion is simple harmonic with amplitude

Options

A. `A`
B. `B`
C. `A+B`
D. `sqrt(A^2+B^2)`

Answer

To solve the problem step by step, we will analyze the motion of the particle described by the equation \( x = A + B \sin(\omega t) \) and determine the amplitude of the motion.

### Step 1: Understand the given equation
The equation of motion is given as:
\[ x = A + B \sin(\omega t) \]
Here, \( A \) is a constant that shifts the motion along the x-axis, and \( B \sin(\omega t) \) represents the oscillatory part of the motion.

**Hint:** Identify the components of the equation: the constant \( A \) and the oscillatory term \( B \sin(\omega t) \).

### Step 2: Determine the initial displacement
At \( t = 0 \):
\[ x(0) = A + B \sin(0) = A + B \cdot 0 = A \]
Thus, the initial displacement \( x_0 \) is equal to \( A \).

**Hint:** Substitute \( t = 0 \) into the equation to find the initial position.

### Step 3: Find the maximum displacement
The maximum value of the sine function is 1. Therefore, the maximum displacement \( x_m \) occurs when \( \sin(\omega t) = 1 \):
\[ x_m = A + B \cdot 1 = A + B \]

**Hint:** Recall that the sine function reaches its maximum value of 1, which will help in finding the maximum displacement.

### Step 4: Calculate the amplitude
The amplitude \( A \) of the simple harmonic motion is defined as the maximum displacement from the mean position. The mean position in this case is represented by the constant \( A \). Thus, the amplitude can be calculated as:
\[ \text{Amplitude} = x_m - x_0 = (A + B) - A = B \]

**Hint:** The amplitude is the difference between the maximum displacement and the initial displacement.

### Step 5: Conclusion
The amplitude of the motion described by the equation \( x = A + B \sin(\omega t) \) is:
\[ \text{Amplitude} = B \]

**Final Answer:** The amplitude of the motion is \( B \).