Two straight tunnels are dug through a uniform Earth: one along a diameter and the other along a chord. A particle is released from one end of each tunnel. Let T₁,T₂ be their periods and v₁,v₂ their maximum speeds for the diameter and chord tunnels respectively. Select the correct statement(s).
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Question

Two straight tunnels are dug through a uniform Earth: one along a diameter and the other along a chord. A particle is released from one end of each tunnel. Let T₁,T₂ be their periods and v₁,v₂ their maximum speeds for the diameter and chord tunnels respectively. Select the correct statement(s).

Select all that apply

G1 Physics · Accepted Answer

Step 1: Inside a uniform Earth, gravitational acceleration is proportional to distance from the centre. Along any straight tunnel, the component along the tunnel is -(g/R)x, where x is displacement from the tunnel midpoint. Step 2: Thus both motions obey x¨+(g/R)x=0 and have the same period T=2π√(R/g), so T₁=T₂. Step 3: Maximum speed is ω times amplitude. The diameter-tunnel amplitude is R, whereas a chord-tunnel amplitude is smaller than R; hence v₁>v₂. Final: A and D.