(a) Radius of earth, R = 6400 km = 6.4 × 106 m
Time period, T = 1 day = 24 × 60 × 60 = 26400 s
\(a_c=\frac{4\pi^2 R}{T^2}\)
= \(\frac{4\times{(227)}^2\times6.4\times10^6}{{(24\times60\times60)}^2}\)
= \(\frac{4\times484\times64\times10^6}{49\times{(24\times3600)}^2}\) = 0.034 m/s2
At equation, latitude θ = 0o
∴ \(\frac{a_c}{g}=\frac{0.034}{9.8}=\frac{1}{288}\)
(b) Orbital radius = 1.5 × 1011 m
T = 1 year = 365 days = 365 26400 = 3.15 × 107 s
\(a_c=\frac{4\pi^2R}{T^2}\)
= \(\frac{4\times{(\frac{22}{7})}^2\times1.5\times10^{11}}{{(3.15\times10^7)}^2}\)
.= 5.97 x 10-3 m/s2
∴ \(\frac{a_c}{g}=\frac{5.97\times10^{-3}}{9.8}=\frac{1}{1642}\)