Question

A massless spring lying on a smooth horizontal surface is compressed by a force of \(63.5 \mathrm{~N}\), which results in a displacement of \(4.35 \mathrm{~cm}\) from the initial equilibrium position. How much work will it take to compress the spring from \(4.35 \mathrm{~cm}\) to \(8.15 \mathrm{~cm} ?\)

Short answer

Answer: 23.52 J

Step-by-step solution

Write down the Hooke's Law formula

Hooke's Law states that the force (F) applied to a spring is directly proportional to its displacement (x) from the equilibrium position, given by the formula \(F = -kx\), where k is the spring constant. Note that the negative sign indicates that the force acts in the direction opposite to the displacement.

Calculate the spring constant (k)

We are given the force F (\(63.5 \mathrm{~N}\)) and the displacement x (\(4.35 \mathrm{~cm}\)). We can use these values to find the spring constant (k) by rearranging the Hooke's Law formula: \(k = -\frac{F}{x}\) Since the spring is compressed, the force acts in the opposite direction to the displacement: \(F = 63.5 \mathrm{~N}\) and \(x = -4.35 \mathrm{~cm}\) (negative sign to indicate the opposite direction). We first convert the displacement to meters: \(x = -0.0435 \mathrm{~m}\). Then, we calculate the spring constant: \(k = -\frac{63.5}{-0.0435} = 1460.92 \mathrm{~N/m}\)

Write down the equation for the potential energy stored in a spring

The potential energy (U) stored in a spring can be calculated using the equation \(U = \frac{1}{2}kx^2\).

Calculate the work needed to further compress the spring

To find the work required to compress the spring from the initial displacement (\(x_1 = -0.0435 \mathrm{~m}\)) to the final displacement (\(x_2 = -0.0815 \mathrm{~m}\)), we will find the difference in potential energy between these two displacements. The work needed will be equal to this difference: \(W = U_2 - U_1 = \frac{1}{2}kx_2^2 - \frac{1}{2}kx_1^2\) Now, we can substitute the values of the spring constant (k) and the displacements (x_1 and x_2) into the equation: \(W = \frac{1}{2}(1460.92)(-0.0815)^2 - \frac{1}{2}(1460.92)(-0.0435)^2\) \(W = 23.52 \mathrm{~J}\) Therefore, it will take \(23.52 \mathrm{~J}\) of work to compress the spring from \(4.35 \mathrm{~cm}\) to \(8.15 \mathrm{~cm}\).

Key concepts

Spring Constant

The spring constant, symbolized by the letter 'k', is a measure of a spring's stiffness. When dealing with problems related to springs, understanding the spring constant is crucial. The spring constant relates the amount of force required to displace a spring from its equilibrium position. In precise terms, Hooke's Law can be written as
\( F = -kx \),
where 'F' denotes the force applied to the spring and 'x' represents the displacement from the spring's original position.

When the spring constant is high, the spring is stiffer, requiring more force to achieve the same displacement. Conversely, a small spring constant indicates a spring that can be stretched or compressed easily. It's also worth noting that the spring constant has units of Newtons per meter (N/m), providing an indication of the force per unit length needed to stretch or compress the spring.

Potential Energy in a Spring

The potential energy stored in a spring is the energy that it holds when it is stretched or compressed. This energy can be calculated using the formula
\( U = \frac{1}{2}kx^2 \),
where 'U' is the potential energy, 'k' is the spring constant, and 'x' is the displacement from the equilibrium position. This expression explains how that energy is directly proportional to the square of the displacement—meaning that doubling the displacement will quadruple the potential energy stored in the spring.

To visualize this, imagine drawing back a slingshot—the further back you pull, the more energy you store and the more powerful the release. Remember, this energy is 'potential' because it can be transformed into other forms of energy, such as kinetic energy, when the spring returns to its equilibrium position.

Work-Energy Principle

The work-energy principle is a fundamental concept in physics which relates the work done by or against forces (other than friction) to the change in the energy of an object. When discussing springs, the principle implies that the work done in stretching or compressing a spring will result in an equivalent amount of potential energy stored within the spring.

Work done (W) is mathematically expressed as the force multiplied by the displacement in the direction of the force:
\( W = Fd \). In the context of a spring, the work done in compressing or stretching the spring is equal to the change in its potential energy:
\( W = U_2 - U_1 \),
where 'W' stands for the work done, 'U_1' is the initial potential energy, and 'U_2' is the final potential energy.

This principle helps us understand that if we know the initial and final states of the spring, we can calculate the work done in moving between these states. The beauty of the work-energy principle is that it is independent of the path taken and depends only on the start and end points. For example, whether you gradually compress a spring or do it quickly, if the compression is the same, the work done—and thus the energy stored—will also be the same.