Explanation Given information: r = 1 θ (1) d θ d t = t 2 (2) Formula used: Write the general expression to calculate the velocity in cylindrical coordinates. v = r ˙ u r + r θ ˙ u θ + z ˙ k (3) Write the general expression to calculate the acceleration in cylindrical coordinates. a = ( r ¨ − r θ ˙ 2 ) u r + ( r θ ¨ + 2 r ˙ θ ˙ ) u θ + z ¨ k (4) Calculation: Since z is not given, the equation (3) and (4) becomes, v = r ˙ u r + r θ ˙ u θ (5) a = ( r ¨ − r θ ˙ 2 ) u r + ( r θ ¨ + 2 r ˙ θ ˙ ) u θ (6) From equation (2), d θ d t = t 2 θ ˙ = t 2 { ∵ x ˙ = d x d t } θ ¨ = 2 t { ∵ x ¨ = d 2 x d t 2 } From equation (1), r = 1 θ r ˙ = − θ ˙ θ 2 (7) r ¨ = ( θ 2 ) ( − θ ¨ ) − ( − θ ˙ ) ( ( 2 θ ) θ ˙ ) ( θ 2 ) 2 { ∵ u v = v u ' − u v ' v 2 } r ¨ = 2 θ ( θ ˙ ) 2 − θ 2 θ ¨ θ 4 (8) Substitute t 2 for θ ˙ in equation (7). r ˙ = − t 2 θ 2 Substitute t 2 for θ ˙ and 2 t for θ ¨ in equation (8). r ¨ = 2 θ ( t 2 ) 2 − θ 2 ( 2 t ) θ 4 = 2 t 4 − 2 t θ θ 3 = 2 t ( t 3 − θ ) θ 3 Substitute t 2 for θ ˙ , 1 θ for r , and − t 2 θ 2 for r ˙ in equation (5) to find the velocity

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Chapter 13.6, Problem 2E

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Calculate the velocity and acceleration vectors in terms of ur and uθ for r=1θ and dθdt=t2.

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Calculate the velocity and acceleration vectors in terms of u r and u θ for r = 1 θ and d θ d t = t 2 .

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Calculate the velocity and acceleration vectors in terms of ur and uθ for r=1θ and dθdt=t2.

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