Q2. A planimeter measures the area of a figure traversed clockwise with the anchor point inside. M = 100 cm², additive constant C = 20, initial reading I = 3.436, final reading F = 8.945, N = −1 (zero mark crossed once in reverse direction). The area measured was:
📚 Detailed Explanation: Planimeter with Anchor Inside — A = 1550.9 cm²
When a planimeter’s anchor point is placed inside the boundary of the figure, the instrument under-reads the actual area by one zero circle. The zero circle is an imaginary circle traced by the tracer arm when the anchor is at the centre and the arm rotates a full 360°. To compensate, the instrument constant C (equal to the zero circle area, expressed in dial units multiplied by M) is added to the formula.
The formula is identical to the standard case, but C is now included. The anchor-inside setup is used when the figure is so large that placing the anchor outside would require an impractically long tracer arm.
Step-by-Step Calculation
M = 100 cm², C = 20, I = 3.436, F = 8.945, N = -1
A = M × [F – I ± 10N + C]
A = 100 × [8.945 – 3.436 – 10(1) + 20] ↠N = -1, so ±10N = -10
A = 100 × [5.509 – 10 + 20]
A = 100 × [15.509]
A = 1550.9 cm²
Why the Other Options Are Wrong
A (1760.8 cm²): Likely from using N = +1 (forward crossing) instead of −1 (reverse). That gives [8.945 − 3.436 + 10 + 20] = 35.509, × 100 = 3550.9. Doesn’t match — an additional arithmetic error also exists here.
B (1108.16 cm²): Possibly from omitting the 10N correction entirely (treating N = 0): [8.945 − 3.436 + 20] = 25.509, × 100 = 2550.9. Still doesn’t match, indicating C was also misread or M was applied incorrectly — a compounded error.
D (1914.7 cm²): No direct standard substitution error produces exactly this value. Likely from confusing F and I (using I − F instead of F − I) along with a sign error in N.
Anchor Inside vs. Outside: At a Glance
| Situation | Formula | Why |
|---|---|---|
| Anchor outside figure (standard) | A = M × [F − I ± 10N] | No zero circle contribution; disc records actual swept area |
| Anchor inside figure | A = M × [F − I ± 10N + C] | Planimeter under-reads by one zero circle area; C corrects this |
Key Concepts for Students
- Zero circle and C: The zero circle is not a physical circle on the instrument — it is the circular path the tracer would trace if the anchor arm rotated 360° without the tracer moving relative to the pivot. Its area, converted to dial units and scaled by M, gives the value of C.
- N = −1 means backward crossing: When the disc’s zero mark crosses the fixed index in the direction opposite to the tracing direction, use N = −1 and subtract 10 in the formula. Forward crossing gives N = +1 and adds 10.
- Practical exam tip: In planimeter numericals, always list the four known values (M, F, I, N, C) before substituting. Confirming the sign of N and whether C is included prevents the two most common errors in these problems.