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Hooke's Law is a fundamental principle that describes how springs behave under different forces. When a spring is either compressed or stretched, it experiences a force proportional to its displacement from the equilibrium position.
The formula for Hooke's Law is expressed as:

• \( F = -kx \)

Here:

• \( F \) is the force applied to the spring (in Newtons)
• \( k \) is the spring constant (a measure of the stiffness of the spring)
• \( x \) is the displacement from the spring's original length (in meters)

The negative sign in the equation indicates that the force exerted by the spring is always in the opposite direction of the displacement. This means the spring tends to return to its original state, pushing or pulling against the force applied to it. When calculating work based on Hooke's Law, it's important to determine the precise displacement and apply the correct spring constant.

The spring constant, denoted as \( k \), is a crucial parameter in understanding how a spring reacts to forces. It defines the stiffness of a spring.
The higher the spring constant, the stiffer the spring, and the harder it is to deform. Conversely, a lower spring constant means the spring is more easily stretched or compressed.
The units for the spring constant are typically expressed in Newtons per meter (N/m). In the given problem, the original spring constant is provided as 3.8 kN/m.
This value signifies that for every meter the spring is compressed or stretched, 3.8 kilonewtons of force is applied. To convert this into a more common measurement, we multiply by 1,000 (since 1 kN = 1,000 N):

• \( 3.8 \text{ kN/m} = 3800 \text{ N/m} \)

For practical calculations and accurate results, converting units to the SI system is crucial. Investigating this conversion allows us to apply the spring constant rightly in applying Hooke's law and calculating the work done on the spring.

Unit conversion is a crucial step in solving physics problems, especially when dealing with measurements in different units. In problems involving work and energy, it is essential to have all quantities in the same set of units.
In the given exercise:

• The spring length is given in centimeters, which needs converting to meters.
• The spring constant is provided in kilonewtons per meter, which should be in newtons per meter.

Let's break down these conversions:

• \( 45 \text{ cm} = 0.45 \text{ m} \)
• \( 32 \text{ cm} = 0.32 \text{ m} \)

This ensures that all lengths are in meters before calculating displacement. For the spring constant:

• \( 3.8 \text{ kN/m} = 3800 \text{ N/m} \)

Converting all relevant quantities to their appropriate units allows us to apply physical laws accurately without encountering errors due to unit mismatches.
These conversion steps help maintain consistency in calculations and are essential in determining the correct amount of work done in compressing the spring.