Work Done by the Gravitational Force — Question 9

Question 9

A shared height-zero error cancels from gravitational work

Take a uniform gravitational field with $g=10$ m/s$^2$, upward y positive, and neglect air resistance. A 1 kg object moves from a displayed height of 5 m to 2 m. Both readings share the same unknown additive offset b in $[-0.2,0.2]$ m: displayed height equals true height plus b.

1. Find the exact gravitational work despite the unknown offset.

2. If instead the two offsets were independent in the same bounds, find the allowed work interval.

3. Why should the common-offset case not be treated by independently maximizing both endpoint errors?

Question 9 - Solution

A shared height-zero error cancels from gravitational work

(a) Subtract before inserting the uncertain zero.

The governing relation is $$\boxed{W_g=mg[(y_{i,\mathrm{display}}-b_i)-(y_{f,\mathrm{display}}-b_f)]}$$

$$y_i-y_f=(5-b)-(2-b)=3\ \mathrm m,$$ so $$W_g=30\ \mathrm J$$ exactly in this model.

(b) Allow independent extrema only for independent offsets.

Now the true drop is $$3-b_i+b_f\in[2.6,3.4]\ \mathrm m.$$ Gravity work is in $[26,34]\ \mathrm J$.

(c) Respect the correlation.

The same b cannot simultaneously take different values at the two endpoints. Independent corner choices introduce states excluded by the common-offset model and would produce an unnecessarily wide interval.