A spring of negligible mass has force constant $k = 1600$ N/m. How far must the spring be compressed for $3.20$ J of potential energy to be stored in it?

Verified step by step guidance

1. Step 1: Recall the formula for the potential energy stored in a spring, which is given by \( U = \frac{1}{2} k x^2 \), where \( U \) is the potential energy, \( k \) is the spring constant, and \( x \) is the compression distance.
2. Step 2: Rearrange the formula to solve for \( x \). Start by multiplying both sides of the equation by 2 to eliminate the fraction: \( 2U = kx^2 \). Then divide both sides by \( k \): \( x^2 = \frac{2U}{k} \). Finally, take the square root of both sides: \( x = \sqrt{\frac{2U}{k}} \).
3. Step 3: Substitute the given values into the formula. The potential energy \( U \) is 3.20 J, and the spring constant \( k \) is 1600 N/m. The equation becomes \( x = \sqrt{\frac{2 \times 3.20}{1600}} \).
4. Step 4: Simplify the expression inside the square root. Calculate \( 2 \times 3.20 \) to find the numerator, and divide it by 1600 to find the fraction.
5. Step 5: Take the square root of the resulting value to determine the compression distance \( x \). This will give the final answer for how far the spring must be compressed.

Key Concepts

Hooke's Law

Hooke's Law states that the force exerted by a spring is directly proportional to its displacement from the equilibrium position, expressed as F = -kx, where F is the force, k is the spring constant, and x is the displacement. This principle is fundamental in understanding how springs behave under compression or extension.

Elastic Potential Energy

Elastic potential energy is the energy stored in a spring when it is compressed or stretched. It can be calculated using the formula U = 1/2 kx², where U is the potential energy, k is the spring constant, and x is the displacement. This concept is crucial for determining how much energy is stored in the spring based on its compression.

Energy Conservation

The principle of energy conservation states that energy cannot be created or destroyed, only transformed from one form to another. In the context of springs, the work done to compress the spring is converted into elastic potential energy, which can later be released as kinetic energy when the spring returns to its equilibrium position.