91Ó°ÊÓ

Hooke's Law fundamentally describes the relationship between the force exerted by a spring and its displacement from the equilibrium position. It is mathematically expressed as \( F = -kx \). Here, \( F \) represents the force by the spring, \( k \) is the spring constant, and \( x \) is the displacement. This negative sign signifies that the spring force acts in the opposite direction of displacement, aiming to restore equilibrium.

When a spring is stretched or compressed, the force it applies is directly proportional to the amount of stretch or compression, characterized by the spring constant \( k \). In our scenario, with \( k = 82.0 \, \mathrm{N/m} \), a displacement of \( +0.120 \, \mathrm{m} \) leads to a force of \( -9.84 \, \mathrm{N} \), indicating that the spring pulls back towards equilibrium in the negative direction.

Angular frequency, a key element in simple harmonic motion, tells us how quickly an object oscillates back and forth. It is represented by \( \omega \) and given by:\[ \omega = \sqrt{\frac{k}{m}} \]

In simple harmonic oscillators like springs, \( \omega \) (in radians per second) helps determine the motion's speed and periodicity. For our block-spring system, substituting the given spring constant \( k = 82.0 \, \mathrm{N/m} \) and mass \( m = 0.750 \, \mathrm{kg} \), results in \( \omega \approx 10.46 \, \mathrm{rad/s} \).

This value of angular frequency implies the block completes its oscillatory movement fairly quickly, making it crucial for evaluating the system's dynamic properties.

Maximum speed of an object in simple harmonic motion is when it passes through its equilibrium position. This maximum speed is captured by the formula:

\[ v_{max} = \omega x_{max} \]

Here, \( v_{max} \) is the maximum speed, \( \omega \) is the angular frequency, and \( x_{max} \) is the maximum displacement. For the spring system, using \( \omega = 10.46 \, \mathrm{rad/s} \) and \( x_{max} = 0.120 \, \mathrm{m} \), we calculate a maximum speed of \( 1.2552 \, \mathrm{m/s} \).

The maximum speed occurs as energy in the spring transfers from potential to kinetic, reaching its peak at this point. This is crucial, as it links to how energy fluctuates in oscillatory systems.

In simple harmonic motion, maximum acceleration occurs where displacement is maximal—at the endpoints of the motion. The formula for maximum acceleration is given as:

\[ a_{max} = \omega^2 x_{max} \]

This relates acceleration not only to the maximum displacement \( x_{max} \), but also to the square of the angular frequency \( \omega \). For our case, with \( \omega = 10.46 \, \mathrm{rad/s} \) and \( x_{max} = 0.120 \, \mathrm{m} \), the maximum acceleration is \( 13.1 \, \mathrm{m/s^2} \).

Understanding maximum acceleration helps us comprehend how forces change as the object undergoes oscillations. It provides insights into the potential stresses and strains exerted within the system at regular intervals.