Spherometer: Radius of Curvature
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flagWhat you'll discover
- arrow_forwardExplain how a spherometer measures tiny vertical heights
- arrow_forwardTake zero and surface readings to find the sagitta h
- arrow_forwardApply R = l²/6h + h/2 with the leg separation l
- arrow_forwardJudge when the tip "just touches" and avoid over-driving the screw
Three legs and a screw
A spherometer stands on three legs forming an equilateral triangle of side l, with a micrometer screw at the exact centre. On a flat surface the tip and the legs all touch at the same level. On a convex watch glass, the centre bulges up by a tiny height h — the sagitta — and the screw must be raised by exactly h to touch.
Geometry of a circle through the three leg points gives the elegant result R = l²/6h + h/2, where R is the radius of the sphere the surface belongs to. A bump of half a millimetre on a 4 cm triangle reveals a radius of half a metre — small heights decode huge curvatures.
Procedure: difference of two readings
First place the instrument on plane glass and turn the screw down until the tip just touches; note reading a from the pitch scale (mm) and disc scale (100 divisions, LC = 0.01 mm). Then move to the watch glass and repeat for reading b. The sagitta is simply h = b − a.
Measure l by pressing the legs on paper and measuring the three pin-prick separations with a ruler or vernier, then averaging. In the simulation the touch is shown by a green glow; in the real lab you watch the gap and the reflection of the tip, or listen for the faint scratch as you slide the instrument.
Errors: the "just touches" problem
The whole experiment hangs on judging contact. Drive the screw too far and you lift the legs, reading h too large; stop early and h is too small. Approach the surface slowly from above, and always rotate in one direction to dodge backlash error in the screw.
Other error sources for your report: the legs may not form a perfect equilateral triangle (measure all three sides), the glass may flex under pressure, and temperature changes alter the screw. Since h appears in the denominator, a 10% error in a tiny h becomes a 10% error in R — h is the quantity to measure most carefully.