A converging lens forms a real image of an object

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Uncertainty in measurement · Uncertainty in measurement

Measurement and Method, Ch Propagating uncertainty into the quantity you actually want

Problem

A converging lens forms a real image of an object. The object distance

p

is read from a steel rule as

p = 300.0\,\mathrm{mm}

with uncertainty

u(p) = 0.5\,\mathrm{mm}

(one standard deviation), and the image distance

q

is read from a vernier scale on the screen's translation stage as

q = 120.0\,\mathrm{mm}

with uncertainty

u(q) = 0.3\,\mathrm{mm}

(one standard deviation), the two readings independent. The focal length follows from the thin-lens formula,

\frac{1}{f} = \frac{1}{p} + \frac{1}{q}, \qquad\text{i.e.}\qquad f = \frac{pq}{p+q},

which is assumed to hold exactly for this lens. ((a)) Find

f

, and then find

u(f)

by moving

p

and

q

in turn by their own uncertainties, recomputing

f

each time, and combining the two changes in quadrature. State

f

with its uncertainty. ((b)) The vernier scale on the screen's stage can be read to half its present uncertainty at no extra cost. A new digital calliper for the object-distance rail would halve

u(p)

but would use a term's equipment budget. Using the contributions found in part (a), decide which improvement is worth making, and estimate the resulting

u(f)

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