A uniform rope of mass M and length R lies along a radius and rotates in a horizontal plane about one end with constant angular speed ω. If x is the distance of a point of the rope from the axis, the tension T(x) at that point is:
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Question

A uniform rope of mass M and length R lies along a radius and rotates in a horizontal plane about one end with constant angular speed ω. If x is the distance of a point of the rope from the axis, the tension T(x) at that point is:

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G1 Physics · Accepted Answer

Let the linear mass density be λ = M/R. Consider a small element of the rope at distance r from the axis, of length dr. Its mass is dm = λdr. The element requires inward centripetal force dF = ω²r dm = λω²r dr. The section of rope at x must supply the centripetal force required by the complete portion from r=x to r=R. Therefore T(x) = ∫[x to R] λω²r dr = λω²(R² − x²)/2. Using λ=M/R, T(x) = Mω²(R² − x²)/(2R).