A linear harmonic oscillator of force constant 2*10^6N/m and amplitude 0.01 m has a total mechanical energy of 160 joules Its

To analyze the linear harmonic oscillator you've described, we can break down the components involved, including the force constant, amplitude, and total mechanical energy. This will help us understand how these factors interact in the system.

Understanding the Components

A linear harmonic oscillator is a system that experiences a restoring force proportional to its displacement from an equilibrium position. The key parameters here are:

• Force Constant (k): This is given as 2 x 10⁶ N/m. It indicates how stiff the spring is; a higher value means a stiffer spring.
• Amplitude (A): The maximum displacement from the equilibrium position, which is 0.01 m in this case.
• Total Mechanical Energy (E): This is the sum of kinetic and potential energy in the system, given as 160 joules.

Energy in a Harmonic Oscillator

The total mechanical energy in a harmonic oscillator can be expressed using the formula:

E = (1/2) k A²

Here, E represents the total mechanical energy, k is the force constant, and A is the amplitude. This formula tells us that the energy is dependent on both the stiffness of the spring and the square of the amplitude.

Calculating Total Energy

Let's plug in the values you provided to see if they align with the total mechanical energy of 160 joules:

E = (1/2) (2 x 10⁶ N/m) (0.01 m)²

Calculating this step-by-step:

• First, calculate A²: (0.01 m)² = 0.0001 m².
• Next, multiply by k: 2 x 10⁶ N/m * 0.0001 m² = 200 N·m.
• Finally, multiply by 1/2: (1/2) * 200 N·m = 100 joules.

From this calculation, we find that the total mechanical energy is actually 100 joules, not 160 joules. This discrepancy suggests that either the amplitude or the force constant may need to be re-evaluated, or that the total energy provided might not correspond to the given parameters.

Exploring Further Implications

In a real-world scenario, if the total mechanical energy is indeed 160 joules, we can rearrange the energy formula to find the necessary amplitude:

A = sqrt((2E)/k)

Substituting the known values:

A = sqrt((2 * 160 J) / (2 x 10⁶ N/m))

A = sqrt(320 J / (2 x 10⁶ N/m)) = sqrt(0.00016 m²) = 0.01265 m.

This means that if the total mechanical energy is indeed 160 joules, the amplitude would need to be approximately 0.01265 m, which is larger than the initially stated amplitude of 0.01 m.

Final Thoughts

In summary, the relationship between force constant, amplitude, and total mechanical energy is crucial for understanding the dynamics of a harmonic oscillator. If the parameters provided do not yield consistent results, it’s essential to reassess them to ensure accurate modeling of the system. This exercise not only reinforces the principles of harmonic motion but also highlights the importance of precision in physical measurements.