Direct method Here, we find an expression for the position of P as a function of time. Then, the velocity and acceleration are obtained by simple differentiation. Since there is no sliding, we have Rcosφi+Rsin Uo- Rosin i+Rφcosφj -wR(l +sin )i+aRcos o j aP=P=ao-R(φsinφ+φcos)i+R(φcosφ-φsino)j -aR(l +sin o)-w2Rcos p]i+arcos o-w2Rsin o]j Relative motion with respect to O Here, we can directly use either set of the expressions given previously, 4 and 5, or, 9 and 10, with vor+w X O+山xrp+×(u×Tp) rp=Rcosφi+Rsin =wk. c=ak R(1+sinφ)i+ aRcosφ aR(1+sin ()-w2Rcos o ]i+ [ aRcos o-w2Rsin g]j Here, we use expressions 4 and 5, or 9 and 10, with O replaced by C r CP u×rCp)Direct Method : Here, we find an expression for the position of P as a function of time. Then, the velocity and acceleration are obtained by simple differentiation. Since there is no sliding, we have, vO′ = −ωR i, aO′ = −αR i, and, rP = rO′ + R cos φ i + R sin φ j . Therefore, vP = r˙ P = vO′ − Rφ˙ sin φ i + Rφ˙ cos φ j = −ωR(1 + sin φ) i + ωR cos φ j aP = r¨P = aO′ − R(φ¨ sin φ + φ˙ cos φ) i + R(φ¨ cos φ − φ˙ sin φ) j = [−αR(1 + sin φ) − ω 2R cos φ] i + [αR cos φ − ω 2R sin φ] j Relative motion with respect to O′ : Here, we can directly use either set of the expressions given previously, 4 and 5, or, 9 and 10, with, vP = vO′ + ω × r ′ P (17) aP = aO′ + ω˙ × r ′ P + ω × (ω × r ′ P ) , (18) r ′ P = R cos φ i + R sin φ j , ω = ωk, α = αk , vO′ = −ωR i , aO′ = −αR i, and k = i × j. Thus, vP = −ωR(1 + sin φ) i + ωR cos φ j aP = [−αR(1 + sin φ) − ω 2R cos φ] i + [αR cos φ − ω 2R sin φ] j . Relative motion with respect to C : Here, we use expressions 4 and 5, or 9 and 10, with O′ replaced by C. vP = ω × r ′ CP (19) aP = aC + ω˙ × r ′ CP + ω × (ω × r ′ CP ) . (20) 6