A nonlinear spring Hooke’s law is applicable to idealized (linear) springs that are not stretched or compressed too far from their equilibrium positions. Consider a nonlinear spring whose restoring force is given by F(x) = 16x−0.1x³, for |x|≤7. 
b. How much work is done in stretching the spring from its equilibrium position (x=0) to x=1.5?

Verified step by step guidance

1. Recall that the work done by a variable force \( F(x) \) in moving an object from position \( a \) to \( b \) is given by the definite integral of the force over that interval: \[ W = \int_{a}^{b} F(x) \, dx \].
2. Identify the limits of integration from the problem: the spring is stretched from the equilibrium position \( x=0 \) to \( x=1.5 \), so \( a=0 \) and \( b=1.5 \).
3. Substitute the given force function \( F(x) = 16x - 0.1x^3 \) into the integral: \[ W = \int_{0}^{1.5} (16x - 0.1x^3) \, dx \].
4. Split the integral into two simpler integrals: \[ W = \int_{0}^{1.5} 16x \, dx - \int_{0}^{1.5} 0.1x^3 \, dx \].
5. Integrate each term using the power rule for integration: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]. So, \[ \int 16x \, dx = 16 \cdot \frac{x^2}{2} = 8x^2 \] and \[ \int 0.1x^3 \, dx = 0.1 \cdot \frac{x^4}{4} = 0.025x^4 \]. Then evaluate these expressions at the limits \( 0 \) and \( 1.5 \) and subtract to find the work done.

Key Concepts

Work Done by a Variable Force

Work done by a force that varies with position is calculated by integrating the force function over the displacement interval. For a spring, this means integrating the restoring force from the initial to the final position to find the total work required to stretch or compress it.

Nonlinear Spring Force

Unlike Hooke’s law for linear springs (F = kx), a nonlinear spring has a restoring force that depends on higher powers of displacement, such as F(x) = 16x − 0.1x³. This means the force changes in a more complex way as the spring stretches, affecting the work calculation.

Definite Integral in Calculus

A definite integral computes the accumulation of quantities, such as work done, over an interval. In this problem, integrating the force function from 0 to 1.5 gives the total work done in stretching the spring, capturing the effect of the nonlinear force.