To determine whether the system is conservative and to find the equivalent velocity when the maximum potential energy is expressed as purely kinetic energy, we need to analyze the given parameters and equations of motion. Let's break this down step by step.
Understanding the System
We have a mass \( m = 2 \, \text{kg} \) and a spring constant \( k \) such that \( \frac{k}{m} = 100 \). From this, we can find \( k \) as follows:
- Given \( \frac{k}{m} = 100 \), we can rearrange it to find \( k = 100m \).
- Substituting \( m = 2 \, \text{kg} \), we get \( k = 100 \times 2 = 200 \, \text{N/m} \).
Analyzing the Motion
The equation of motion is given as:
x(t) = 3 \sin(8t + 30)
This represents a harmonic oscillator, where:
- The amplitude \( A = 3 \, \text{m} \).
- The angular frequency \( \omega = 8 \, \text{rad/s} \).
Determining Conservativeness
A system is considered conservative if the total mechanical energy (kinetic + potential) remains constant over time. In a simple harmonic oscillator, the potential energy \( U \) and kinetic energy \( K \) can be expressed as:
- Potential Energy: \( U = \frac{1}{2} k x^2 \)
- Kinetic Energy: \( K = \frac{1}{2} m v^2 \)
For a conservative system, the total energy \( E \) is constant:
E = K + U
However, in this case, we need to check if the energy is conserved. The maximum potential energy occurs when \( x \) is at its maximum amplitude:
U_{max} = \frac{1}{2} k A^2 = \frac{1}{2} \times 200 \times 3^2 = 900 \, \text{J}
The maximum kinetic energy occurs when the displacement \( x \) is zero:
K_{max} = \frac{1}{2} m v_{max}^2
Setting \( K_{max} = U_{max} \) gives:
900 = \frac{1}{2} \times 2 \times v_{max}^2
Solving for \( v_{max} \):
- Multiply both sides by 2: \( 1800 = 2 v_{max}^2 \)
- Divide by 2: \( 900 = v_{max}^2 \)
- Taking the square root: \( v_{max} = 30 \, \text{m/s} \)
Conclusion on Conservativeness
To conclude, if energy is not conserved due to external forces or damping (which we have not considered here), the system is not conservative. In this case, since we derived a maximum kinetic energy equal to the maximum potential energy, it suggests that under ideal conditions, the system behaves conservatively. However, if there are any dissipative forces (like friction or air resistance), the system would not be conservative.
In summary, while the calculations indicate a conservative behavior under ideal conditions, real-world factors could lead to non-conservative behavior. The equivalent velocity when maximum potential energy is expressed as purely kinetic energy is \( 30 \, \text{m/s} \).
