Every university introductory physics course considers the problem of Atwood's machine taking into account the mass of the pulley. In the usual treatment, the tensions at the two ends of the string are offhandedly taken to act on the pulley and be responsible for its rotation. However, such a free-body diagram of the forces on the pulley is not a priori justified, inducing students to construct wrong hypotheses such as that the string transfers its tension to the pulley or that some symmetry is in operation. We reexamine this problem by integrating the contact forces between each element of the string and the pulley and show that although the pulley does behave as if the tensions were acting on its ends, this comes only as the final result of a detailed analysis. We also address the question of how much friction is needed to prevent the string from slipping over the pulley. Finally, we deal with the case in which the string is on the verge of sliding and show that this cannot happen unless certain conditions are met by the coefficient of static friction and the masses involved.

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16.

It has been noticed by Lubarda (Ref. 17) that Eq. (7) does not define n correctly if Δθ is finite, in particular n is not necessarily parallel to r̂(θ). Let dn/dθ=ñ(θ)r̂(θ) be the normal force per unit angle, with ñ(θ) a continuous function. The normal force on the element of string that subtends the small but finite angle Δθ is n=θΔθ/2θ+Δθ/2ñ(θ)r̂(θ)dθ. By the mean value theorem for integrals, n=(ñ(θ1)cosθ1x̂+ñ(θ2)sinθ2ŷ)Δθ where the angles θ1 and θ2 belong to the closed interval [θΔθ/2,θ+Δθ/2]. Since ñ(θ1)cosθ1=ñ(θ)cosθ+ϵ1 and ñ(θ2)sinθ2=ñ(θ)sinθ+ϵ2, where both ϵ1 and ϵ2 tend to zero as Δθ → 0, it follows that n=ñ(θ)r̂(θ)Δθ+ϵΔθ, where ϵ=ϵ1x̂+ϵ2ŷ. Inserting this into Eq. (8b), dividing by Δθ and letting Δθ → 0 one gets Eq. (13) because limΔθ0ϵ·r̂(θ)=0. The same argument applies to the tangential force f and the corresponding Eq. (12). All we have assumed is that the tangential and normal forces per unit angle, namely, df/ and dn/, are well defined and continuous, which is consistent with Eqs. (12) and (13) that follow from the assumption. Equation (7) is the usual shortcut employed by physicists to avoid the rigorous but lengthy discussion above, which has no effect on the final result. Therefore, our derivation does not suffer from the drawback pointed out by Lubarda.

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