91Ó°ÊÓ

Hooke's Law is a fundamental principle that describes how springs work. It states that the force the spring exerts is directly proportional to the distance it is stretched or compressed from its natural length. The mathematical expression of Hooke's Law is given by:

• \( F = -kx \)

Here, \( F \) represents the force exerted by the spring, \( k \) is the spring constant, and \( x \) is the displacement from the equilibrium position. The negative sign indicates that the force exerted by the spring acts in the opposite direction to the displacement.
For example, if a spring with a spring constant \( k = 82.0 \, \mathrm{N/m} \) is stretched by \( x = 0.120 \, \mathrm{m} \), the force exerted by the spring would be:

• \( F = -82.0 \times 0.120 = -9.84 \, \mathrm{N} \)

This means the spring pulls back with a force of 9.84 Newtons, opposite to the direction in which it was stretched.

Angular frequency \( \omega \) is a measure of how quickly an oscillating system cycles through its motion. It's particularly useful in describing oscillations due to its relation to circular motion. The angular frequency of a mass-spring system is defined by the formula:

• \( \omega = \sqrt{\frac{k}{m}} \)

Where \( k \) is the spring constant and \( m \) is the mass attached to the spring. In our exercise, substituting \( k = 82.0 \, \mathrm{N/m} \) and \( m = 0.750 \, \mathrm{kg} \), we find:

• \( \omega = \sqrt{\frac{82.0}{0.750}} \approx 10.44 \, \mathrm{rad/s} \)

The angular frequency helps us understand how tight or loose the spring system is, with higher values indicating faster oscillations.

The maximum speed in simple harmonic motion is reached as the moving object passes through the equilibrium point. This speed, denoted as \( v_{max} \), can be calculated using the formula:

• \( v_{max} = \omega A \)

Where \( \omega \) is the angular frequency and \( A \) is the amplitude of the motion. From the earlier steps, we have \( \omega \approx 10.44 \, \mathrm{rad/s} \) and amplitude \( A = 0.120 \, \mathrm{m} \). By substituting these values into the equation, we find:

• \( v_{max} = 10.44 \times 0.120 \approx 1.25 \, \mathrm{m/s} \)

This speed is the maximum the block will achieve as it oscillates back and forth in the system.

The maximum acceleration of an object in simple harmonic motion occurs at the endpoints of its path, where it changes direction. The maximum acceleration \( a_{max} \) is determined by:

• \( a_{max} = \omega^2 A \)

Where \( \omega \) is the angular frequency and \( A \) is the amplitude. With \( \omega \approx 10.44 \, \mathrm{rad/s} \) and \( A = 0.120 \, \mathrm{m} \), let's substitute these into the formula:

• \( a_{max} = (10.44)^2 \times 0.120 \approx 13.08 \, \mathrm{m/s^2} \)

This maximum acceleration provides insight into how forcefully the spring system accelerates the block as it moves through its oscillatory motion.