Question

Determine the work required to deflect a linear spring with a spring constant of \(70 \mathrm{kN} / \mathrm{m}\) by \(20 \mathrm{cm}\) from its rest position.

Short answer

Answer: The work required to deflect the linear spring by 20 cm from its rest position is 1.4 kN⋅m.

Step-by-step solution

Convert deflection distance into meters

We are given the deflection distance in centimeters, which we need to convert into meters: \(20 \ \mathrm{cm} = 20 \times 10^{-2} \ \mathrm{m} = 0.2 \ \mathrm{m}\)

Determine the force using Hooke's Law

Hooke's Law states that \(F = kx\). Given the spring constant \(k = 70 \ \mathrm{kN/m}\) and the deflection distance \(x = 0.2 \ \mathrm{m}\), we can calculate the force required to deflect the spring: \(F = kx = 70 \times 0.2 = 14 \ \mathrm{kN}\)

Calculate the work done on the spring

The work done on a spring is given by the formula \(W = \frac{1}{2}kx^2\). Using the spring constant \(k = 70 \ \mathrm{kN/m}\) and the deflection distance \(x = 0.2 \ \mathrm{m}\): \(W = \frac{1}{2}kx^2 = \frac{1}{2} \times 70 \times (0.2)^2 = 1.4 \ \mathrm{kN \cdot m}\) The work required to deflect the linear spring by \(20 \ \mathrm{cm}\) from its rest position is \(1.4 \ \mathrm{kN \cdot m}\).

Key concepts

Understanding Hooke's Law

Hooke's Law is a principle of physics that relates the amount of force exerted on a spring to the distance it is stretched or compressed. The law is commonly expressed through the formula:
\( F = kx \),
where \( F \) represents the force applied to the spring, \( k \) is the spring constant, and \( x \) is the displacement of the spring from its equilibrium position, measured in the same direction as the force. The spring constant \( k \) is a measure of the spring's stiffness; it indicates how much force is needed to stretch or compress the spring by a unit of distance. A higher value of \( k \) means a stiffer spring, requiring more force to achieve the same displacement compared to a spring with a lower spring constant. In the exercise, we see the application of Hooke's Law to determine the force required to compress a spring given its spring constant and the desired displacement.

Calculating Work Done on a Spring

When a force is exerted on a spring, causing it to compress or stretch, work is done on the spring. The work done, in the context of springs, is the energy transferred to the spring as it compresses or stretches. To calculate this work using physics concepts, we employ the work-energy principle.
The work done on a spring is determined using the formula:
\( W = \frac{1}{2}kx^2 \),
where \( W \) represents the work done expressed in joules (or kilonewton-meters for large magnitudes), \( k \) is the spring constant, and \( x \) is the displacement of the spring from its rest position. This formula takes into account that a spring's resistance changes linearly with the displacement based on Hooke's Law. The factor of \( \frac{1}{2} \) accounts for the fact that the force on the spring is not constant but varies from zero at the equilibrium position to its maximum value at the maximum displacement.

Explaining Spring Deflection

Spring deflection refers to the distance a spring is displaced from its rest position due to a force applied to it. It's an important property when working with mechanical systems, as it determines how much a spring will stretch or compress under a specific load. This displacement is directly connected with how springs store potential energy; when a spring is deflected, it stores energy that can be released when the force is removed, returning the spring back to its original position.
Understanding deflection is crucial in designing systems where springs are used for absorbing energy, like in shock absorbers, or for storing energy, such as in a mechanical clock. Handling spring deflection correctly ensures that mechanical components can safely and reliably undergo repeated loading and unloading cycles without material fatigue or failure. In our exercise, the deflection of 20 cm is the change in spring length from its natural, unstressed length.