A spring of negligible mass has force constant k = 1600 N/m. (a) 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. Identify the formula for 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 displacement of the spring from its equilibrium position.
2. Substitute the given values into the formula. Here, $U = 3.20 \text{ J}$ and $k = 1600 \text{ N/m}$. The equation becomes $3.20 = \frac{1}{2} \times 1600 \times x^2$.
3. Simplify the equation to isolate $x^2$. This involves multiplying both sides of the equation by 2 and then dividing by the spring constant $k$.
4. Take the square root of both sides of the equation to solve for $x$, which represents the compression distance of the spring.
5. Check the units and make sure they are consistent throughout the calculation to ensure the accuracy of the result.

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 converts into elastic potential energy, which can later be released as kinetic energy when the spring returns to its equilibrium position.