A motorcycle crosses a convex circular overbridge of radius R and total arc length L at constant speed. Find: (a) the maximum speed for contact at the highest point; (b) if speed is 1/√2 times that value, the distance from the top where contact is lost; (c) the maximum uniform speed that maintains contact over the entire bridge.
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Question

A motorcycle crosses a convex circular overbridge of radius R and total arc length L at constant speed. Find: (a) the maximum speed for contact at the highest point; (b) if speed is 1/√2 times that value, the distance from the top where contact is lost; (c) the maximum uniform speed that maintains contact over the entire bridge.

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G1 Physics · Accepted Answer

At the top, N=mg-mv²/R≥0 gives v(max)=√(Rg). At v=√(Rg/2), loss of contact occurs when N=0: mg cosθ=mv²/R, so cosθ=1/2 and θ=π/3, giving arc distance πR/3. For the entire bridge, the limiting point is the end at angle L/(2R), so v≤√[gR cos(L/(2R))].