91Ó°ÊÓ

In simple harmonic motion, the natural frequency represents how fast an object oscillates without external forces. It's the constant rate at which the system moves back and forth. For a system with a mass \(m\) and a spring constant \(k\), the natural frequency, often expressed as angular frequency \(\omega\), is determined by the equation:\[ \omega = \sqrt{\frac{k}{m}} \]For our given values, where the mass \(m\) of the glider is 0.500 kg, and the spring constant \(k\) is 450 N/m, we find:\[ \omega = \sqrt{\frac{450}{0.500}} = 30 \, \text{rad/s} \]This calculated natural frequency, 30 rad/s, defines the rate at which the glider will naturally oscillate on the air track when disturbed by an external force. Understanding the natural frequency helps explain the dynamic behavior of the system and its stability under different conditions.

The maximum speed in simple harmonic motion occurs when the oscillating object passes through the equilibrium position, where the energy is entirely kinetic. This speed is determined by the product of the natural frequency and the amplitude of motion. The formula to find the maximum speed \(v_{max}\) of the glider is:\[ v_{max} = \omega A \]Here, \(A\) is 0.040 m (the amplitude), and we've calculated \(\omega\) as 30 rad/s. Substituting these values gives:\[ v_{max} = 30 \times 0.040 = 1.2 \, \text{m/s} \]This result shows the fastest speed the glider achieves as it moves back and forth. In practical terms, this speed indicates how quickly the system can respond to disturbances.

Mechanical energy in simple harmonic motion comprises two parts: potential energy stored in the spring and kinetic energy of the moving object. The total mechanical energy \(E\) is conserved and constant throughout the motion. It is calculated using:\[ E = \frac{1}{2}kA^2 \]With \(k\) being the spring constant (450 N/m) and \(A\) the amplitude (0.040 m), we find:\[ E = \frac{1}{2} \times 450 \times 0.040^2 = 0.36 \, \text{J} \]This constant energy signifies that the sum of kinetic and potential energies remains unchanged throughout. At the maximum amplitude, energy is entirely potential; at equilibrium, it's entirely kinetic. This energy conservation ensures that the system continues its oscillation perpetually (assuming no dissipative forces).

During oscillations, acceleration is maximized when the restoring force pulls or pushes the object to its peak displacement. The object's maximum acceleration \(a_{max}\) can be calculated using:\[ a_{max} = \omega^2 A \]Substituting \(\omega = 30 \, \text{rad/s}\) and \(A = 0.040 \, \text{m}\), we obtain:\[ a_{max} = 30^2 \times 0.040 = 36 \, \text{m/s}^2 \]This represents the highest acceleration experienced by the glider as it is furthest from its equilibrium position. Understanding this concept is crucial for analyzing how quickly the object can change its velocity.