To determine the radius of curvature of a given spherical surface by a Spherometer.
A spherometer is a measuring instrument used to measure the radius of curvature of a spherical surface and a very small thickness.
Figure 3.1 is a schematic diagram of a single disk spherometer. It consists of a central leg OS, which can be raised or lowered through a threaded hole V (nut) at the centre of the frame F. The metallic triangular frame F supported on three legs of equal length A, B and C. The lower tips of the legs form three corners of an equilateral triangle ABC and lie on the periphery of a base circle of known radius, r. The lower tip of the central screw, when lowered to the plane (formed by the tips of legs A, B and C) touches the centre of triangle ABC. A circular scale (disc) D is attached to the screw. The circular scale may have 50 or 100 divisions engraved on it. A vertical scale P marked in millimetres or half-millimetres, called main scale or pitch scale P is also fixed parallel to the central screw, at one end of the frame F. This scale is kept very close to the rim of disc D but it does not touch the disc D. This scale reads the vertical distance when the central leg moves through the hole V.
Pitch of a Spherometer
The vertical distance moved by the screw S in one complete rotation of the circular Scale/Disc D is called the pitch (p) of the spherometer. To find the pitch, give full rotation to the screw (say 4 times) and note the distance (d) advanced over the pitch scale.
If the distance d is 4 mm The pitch can be represented as,
Least Count of the Screw Gauge
The Least count (LC) is the distance moved by the spherometer screw, when the screw is turned through 1 division on the circular. We are using a spherometer which has 100 divisions (N) on the disc. The least count can be calculated using the formula,
The formula for the radius of curvature of a spherical surface
From the figure 3.3, O is the centre of the circle. OE = OA = R, radius of the circle. F is the tip of the screw at the same plane with A, B and C. EF = h, AF = a and ∠AFO =
Therefore, geometrically we can write,
OA2 = OF2 + FA2
or, R2 = (R-h)2 + a2
= R2 -2.R.h + h2 + a2
∴ R = (h2 + a2 )/2h
Now, let l be the distance between any two legs of the spherometer as shown in figure 3.6, then from geometry we have, a = . Thus the radius of curvature of the spherical surface can be given by,
∴ R = ( 3h2 + l2 )/6h
From the figure 3.4, the circle is passing through A and C. O is the centre of the circle. OE =R, radius of the circle. F is the tip of the screw at the same plane with A, B and C. CF = h, AF = a ∠EAC = 900.
∴ CE2 = AE2 + AC2
or, (2R2) = (AF2 + FE2) + (CF2 + AF2)
= a2 + (2R -h)2 + h2 + a2
∴ R= a2/2h+ h/2
Now, let l be the distance between any two legs of the spherometer or the side of the equilateral triangle ABC (Fig. 3.4), then from geometry we have, a = l/√3. Thus the radius of curvature of the spherical surface can be given by,
R = a2/2h+ h/2
or, R = l2/6h+ h/2
Least count of spherometer :
Total number of divisions is the in circular scale, N = _______
One linear scale division, L.S.D. = ____ mm
Distance moved by the screw for 4 rotations, d = ________ mm
Pitch of the screw, p = 4/d = ____mm
Therefore, Least Count, L.C. = p/N= ______________mm
Distance between two legs of spherometer:
AB = _______ cm, BC = _______cm, CA = _________cm
∴ l = (AB + BC+ CA)/3 = _______________________cm
Table 3.1 Table for height (h)
Mean value of sagitta, h = _________________ mm = ________________cm
Radius of curvature of the given convex surface, R = (3h2 + l2 )/6h =…………………………..cm
or, Radius of curvature of the given convex surface, R = l2/6h+ h/2 =…………………………… cm
Your may checkout our blog on SCREW GAUGE & LEAST COUNT
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