A coin is pushed down tangentially from a position θ on a cylindrical surface, with a velocity v as shown in Fig. If the coefficient of friction between the coin and surface is μ, find the tangential acceleration of the coin.
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Solution
As the coin slides down, friction becomes kinetic and acts up along the plane. Aaprt from this, this coin experiences mg↓ and N↑ as shown in Fig. Force equation in radial direction: ∑Fr=N−mgcosθ=mar (i) ∑Ft=mgsinθ−fk=ma1 (ii) Law of kinetic friction: fk=μN (iii) We know radial acceleration of the particle ar=mv2R (iv) Substititing fk from Eq. (iii), and ar from Eq. (iv) in (ii), we have at=gsinθ−μNm Now, substituting N from Eq. (i) in Eq. (iv), we have qt=g(sinθ−μcosθ)−μmv2R