Step-by-step explanation:
To find the vial radius of curvature and the angle subtended by one division, we can use the formula:
Radius of Curvature (R) = (L^2) / (8S)
where:
- L is the sight length (in meters)
- S is the distance the bubble is moved off center (in meters)
(a) Vial Radius of Curvature:
Using the given values, L = 100 m and S = 4 divisions × 2 mm/division = 8 mm = 0.008 m.
Substituting these values into the formula:
R = (100^2) / (8 × 0.008) = 125000 m
Therefore, the vial radius of curvature is 125,000 meters.
(b) Angle Subtended by One Division:
To find the angle subtended by one division, we can use the formula:
Angle (θ) = 2πR / (L × 3600)
Using the value of R calculated in part (a) and L = 100 m:
θ = (2π × 125000) / (100 × 3600) ≈ 0.001745 radians
To convert the angle to seconds, we can use the fact that 1 radian is equal to 206,265 seconds:
θ_seconds = 0.001745 × 206265 ≈ 359 seconds
Therefore, one division subtends an angle of approximately 359 seconds.
In summary:
(a) The vial radius of curvature is 125,000 meters.
(b) One division subtends an angle of approximately 359 seconds.