TangentialAndNormalComponents

TangentialAndNormalComponents - Example: Tangential and...

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Unformatted text preview: Example: Tangential and Normal Components of Acceleration Find the decomposition of the acceleration at t = 1 where r(t) = t−1 , ln t, t2 Also, find the unit vectors T and N. Solution We must decompose a(1) = r (1) in the form a(1) = v (1)T(1) + κ(1)v 2 (1)N(1) Step 1: Compute velocity and acceleration: Therefore v(t) = −t−2 , t−1 , 2t , v(1) = −1, 1, 2 , a(t) = 2t−3 , −t−1 , 2 , 1 T(1) = √ −1, 1, 2 , 6 a(1) = 2, −1, 2 Step 2: Compute the speed and its derivative: v (t) = ||r (t)|| = (t−4 + t−2 + 4t2 )1/2 , v (1) = √ 6 −4 − 2 + 8 −4t−5 − 2t−3 + 8t 1 √ , v (1) = =√ −4 + t−2 + 4t2 )1/2 2(t 26 6 Therefore 1 1 1 √ −1, 1, 2 + κv 2 N(1) = −1, 1, 2 + κv 2 N(1) a(1) = √ 6 6 6 Step 3: Compute the normal part κv 2 N(1): 1 1 1 κv 2 N = a(1) − −1, 1, 2 = 2, −1, 2 − −1, 1, 2 = 13, −7, 10 6 6 6 Step 4: Write down the decomposition v (t) = tangential acc a(1) = normal acc. 1 1 −1, 1, 2 + 13, −7, 10 6 6 Step 5: Find N by dividing the normal part 13, −7, 10 by its length √ 318: 1 N(1) = √ 13, −7, 10 318 √ Questions: How do we know that κ(1)v (1)2 = 318/6? What is κ(1)? ...
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