An object of mass 1 kg executes simple harmonic oscillations along the x - axis with a frequency of `(2)/(pi)Hz.` At the position x = 1 m, the object has a kinetic energy of 24 J and potential energy is 8J. The amplitude of the oscillation is

Options

A. `(3)/(2)m`
B. 2m
C. 4m
D. 8m

Answer

To find the amplitude of the simple harmonic motion (SHM) of the object, we can follow these steps:

### Step 1: Calculate Total Energy
The total energy (E) in a simple harmonic motion is the sum of kinetic energy (KE) and potential energy (PE). Given:
- Kinetic Energy (KE) = 24 J
- Potential Energy (PE) = 8 J

Total Energy (E) = KE + PE
E = 24 J + 8 J = 32 J

### Step 2: Find Angular Frequency
The angular frequency (ω) can be calculated from the frequency (f) using the formula:
ω = 2πf

Given the frequency f = (2/π) Hz:
ω = 2π * (2/π) = 4 rad/s

### Step 3: Use Total Energy Formula
The total energy in SHM can also be expressed as:
E = (1/2) * m * ω² * A²

Where:
- m = mass of the object = 1 kg
- ω = angular frequency = 4 rad/s
- A = amplitude (which we need to find)

Substituting the known values into the equation:
32 J = (1/2) * 1 kg * (4 rad/s)² * A²

### Step 4: Simplify the Equation
Now, simplify the equation:
32 J = (1/2) * 1 * 16 * A²
32 = 8 * A²

### Step 5: Solve for Amplitude
Now, isolate A²:
A² = 32 / 8
A² = 4

Taking the square root to find A:
A = √4 = 2 m

### Conclusion
The amplitude of the oscillation is **2 meters**.

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