The spring in a retractable ballpoint pen is 1.8 cm long, with a 300 N / m spring constant. When

step 1 for a spring mass system. Let us suppose we have a spring, a mass is attached to the spring, the

The spring in a retractable ballpoint pen is 1.8 cm long,...

The spring in a retractable ballpoint pen is $1.8 \mathrm{cm}$ long, with a $300 \mathrm{N} / \mathrm{m}$ spring constant. When the pen is retracted, the spring is compressed by $1.0 \mathrm{mm} .$ When you click the button to extend the pen, you compress the spring by an additional $6.0 \mathrm{mm}$. How much energy is required to extend the pen?

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Key Concepts

Hooke's Law

A fundamental principle stating that the force exerted by a spring is directly proportional to its displacement from its equilibrium position. This relationship, expressed as F = kx where k is the spring constant and x is the displacement, is crucial in understanding how springs respond to forces and how energy is stored in them.

Elastic Potential Energy

The energy stored in a spring when it is compressed or extended is known as elastic potential energy. It is given by the formula U = ½ k x², where k is the spring constant and x is the displacement from the spring’s equilibrium position. This concept is essential for calculating the work done in deforming the spring.

Energy Difference Calculation

When a spring is further compressed or extended from an initial state, the additional energy required can be found by calculating the difference in elastic potential energy between the initial and final states. This approach uses the change in the ½ k x² values to determine the work done in producing the incremental displacement.

Unit Conversion

Accurate physics calculations require consistent units across all quantities involved. In problems dealing with spring energy, converting measurements (such as millimeters to meters) ensures that the computed energy values are correct and meaningful.

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