Work Done by a General Variable Force — Question 8

Question 8

Monotone data bound work without choosing an interpolation

A nondecreasing force is measured as 2 N at x=0, 4 N at x=1 m, and 8 N at x=2 m. No functional form between samples is known.

1. Find guaranteed lower and upper bounds on work from 0 to 2 m.

2. Find the work if linear interpolation is additionally assumed.

3. Explain why the interpolation result is not determined by the three measurements alone.

Question 8 - Solution

Monotone data bound work without choosing an interpolation

(a) Bound force on each unit interval.

The governing relation is $$\boxed{F(x_j)\Delta x_j\leq\int_{x_j}^{x_{j+1}}F(x)\,dx\leq F(x_{j+1})\Delta x_j}$$ On [0,1], force lies between 2 and 4 N; on [1,2], it lies between 4 and 8 N. Therefore $$6\le W\le12\ \mathrm J.$$ These are sharp bounds for bounded monotone functions; continuous curves can approach them.

(b) Add two trapezoid areas.

$$W_{\mathrm{linear}}=\tfrac12(2+4)(1)+\tfrac12(4+8)(1)=9\ \mathrm J.$$

(c) Identify the extra model assumption.

Monotonicity allows rises concentrated near either end of each interval. Those alternatives share the sampled values while changing the area. Straight connections select one permissible history rather than establish a unique measured work.