Work Done by a General Variable Force — Question 2

Question 2

A curved force requires more than endpoint averaging

A force along x is $F_x(x)=3x^2$ N when x is inserted in meters. A particle travels from x=0 to x=2 m.

1. Find the exact work and state the units of the coefficient 3.

2. Estimate work using the average of the two endpoint forces.

3. Explain geometrically why that estimate differs from the exact integral.

Question 2 - Solution

A curved force requires more than endpoint averaging

(a) Integrate the polynomial.

The governing relation is $$\boxed{W=\int_{x_i}^{x_f}F_x(x)\,dx,\qquad [3]=\mathrm{N/m^2}}$$ The coefficient has units N/m$^2$. $$W=\int_0^2 3x^2\,dx=[x^3]_0^2=8\ \mathrm J.$$

(b) Evaluate the endpoint trapezoid.

The endpoint forces are 0 and 12 N, giving $$W_{\mathrm{trap}}=\tfrac12(0+12)(2)=12\ \mathrm J,$$ 50 percent above the exact result.

(c) Compare the chord and the curve.

The upward-curving parabola lies below its straight endpoint chord on this interval. The trapezoid therefore includes excess area. Endpoint averaging is exact for a linear force, not for a general force.