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Two tunnels are drilled through a uniform-density Earth. Tunnel T1 passes through the center of the Earth; tunnel T2 makes an angle of 60 degrees with T1 (i.e., T2 is a chord, not a diameter). A particle is placed in each tunnel and allowed to oscillate. Find the ratio T1/T2 of their time periods, assuming no dissipation.
- 1/2
- 1/sqrt(2)
- sqrt(2)
- 1
Correct answer: 1
Solution
For diametric tunnel T1: restoring force = -mg*x/R where x is displacement from center. omega1² = g/R, T1 = 2*pi*sqrt(R/g). For chord tunnel T2: let the perpendicular from Earth's center to the tunnel have length d = R*sin(60 deg) = R*sqrt(3)/2. At displacement s from the midpoint of T2, the particle's distance from Earth's center is r = sqrt(d² + s²). Gravitational force magnitude = m*g*r/R = m*g*sqrt(d²+s²)/R (directed toward center). The component along the tunnel toward midpoint = m*g*sqrt(d²+s²)/R * (s/sqrt(d²+s²)) = m*g*s/R. This is the same restoring force as for T1! Therefore omega2² = g/R, T2 = 2*pi*sqrt(R/g) = T1. Ratio T1/T2 = 1.
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