91Ó°ÊÓ

Hooke's Law is a fundamental principle in the study of elasticity. It describes the behavior of springs and elastic objects. According to Hooke's Law, the force exerted by a spring is directly proportional to the amount it is stretched or compressed. This relationship can be expressed with the formula:

• \( F = kx \)

where \( F \) is the force applied, \( k \) is the spring constant, and \( x \) is the displacement of the spring from its equilibrium position.

This means if you hang a weight from a spring, and if no other forces like air resistance play a part, the spring will stretch a certain distance until the force from the spring exactly balances the weight's force (due to gravity). In our example, when a 1.3 kg block is hung from the spring, it stretches by 9.6 cm. The force here is simply the weight of the block (\( mg \)), which helps us find the spring constant \( k \).

The spring constant, denoted as \( k \), is a measure of a spring's stiffness. It tells us how much force is required to stretch or compress a spring by a unit length. A higher \( k \) value means a stiffer spring. To calculate \( k \) in our exercise, we use Hooke's Law and the information given:

• The force \( F \) is due to gravity: \( mg \) where \( m = 1.3 \) kg and \( g = 9.8 \) m/s².
• The displacement \( x = 0.096 \) m.

By rearranging the formula \( F = kx \), we solve for \( k \):

• \( k = \frac{mg}{x} = \frac{1.3 \times 9.8}{0.096} \approx 132.8125 \text{ N/m} \)

This calculation shows us the stiffness of the spring being used. A larger spring constant indicates the spring will not stretch as easily, which can affect the motion of the block attached to it.

In simple harmonic motion (SHM), the frequency and period are key characteristics. **Frequency** \( f \) is the number of oscillations per second and is measured in Hertz (Hz). **Period** \( T \) is the time taken for one complete cycle of motion and is measured in seconds. These two are related by:

• \( T = \frac{1}{f} \)
• \( f = \frac{1}{T} \)

To find the period in our problem, we use the formula:

• \( T = 2\pi \sqrt{\frac{m}{k}} \)

where \( m = 1.3 \text{ kg} \) and \( k = 132.813 \text{ N/m} \). This resulted in a period of about 0.627 seconds. The frequency is simply the reciprocal, \( f \approx 1.595 \text{ Hz} \), meaning the spring-block system completes about 1.595 cycles each second.

Amplitude in simple harmonic motion (SHM) is the maximum extent of a vibration or oscillation, measured from the position of equilibrium. In the exercise, the block was initially stretched by 9.6 cm and then pulled an additional 5.0 cm before being released. Thus, the total amplitude \( A \) of the motion is the sum of these displacements:

• \( A = 0.096 + 0.05 = 0.146 \text{ m} \)

The **maximum speed** \( v_{max} \) of the block during SHM occurs as it passes through the equilibrium position. This speed can be calculated using:

• \( v_{max} = A\sqrt{\frac{k}{m}} \)

Substituting the amplitude \( A = 0.146 \text{ m} \), spring constant \( k = 132.813 \text{ N/m} \), and mass \( m = 1.3 \text{ kg} \), the maximum speed \( v_{max} \approx 1.688 \text{ m/s} \). This tells us how fast the block moves at its quickest point, which is critical for understanding the dynamics of the SHM.