To determine Find the acceleration vector a in the form a = a T T + a N N without finding T and N for the given vector r ( t ) = ( a cos t ) i + ( a sin t ) j + b t k . Explanation Given information: r ( t ) = r = ( a cos t ) i + ( a sin t ) j + b t k (1) Formula used: Write the general expression to calculate the velocity. v = d r d t (2) Here, d r d t is the rate of change in displacement with respect to time. Write the general expression to calculate the acceleration. a = d v d t (3) Here, d v d t is the rate of change in velocity with respect to time. Write the general expression for the acceleration vector. a = a T T + a N N (4) Take magnitude on both sides of the equation (4). | a | = a T 2 + a N 2 (5) Write the general expression to calculate the tangential scalar component of acceleration. a T = d d t | v | (6) Calculation: Substitute equation (1) in equation (2). v = d d t ( ( a cos t ) i + ( a sin t ) j + b t k ) v = ( − a sin t ) i + ( a cos t ) j + b k (7) Take magnitude on both sides of the above equation. | v | = ( − a sin t ) 2 + ( a cos t ) 2 + b 2 = a 2 sin 2 t + a 2 cos 2 t + b 2 = a 2 ( sin 2 t + cos 2 t ) + b 2 | v | = a 2 + b 2 { ∵ sin 2 θ + cos 2 θ = 1 } (8) Substitute equation (8) in equation (6). a T = d d t ( a 2 + b 2 ) = 0 Substitute equation (7) in equation (3)

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ISBN: 9780134438986

Author: Joel R. Hass, Christopher E. Heil, Maurice D. Weir

Publisher: PEARSON

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Chapter 13.5, Problem 1E

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Find the acceleration vector a in the form a=aTT+aNN without finding T and N for the given vector r(t)=(acost)i+(asint)j+btk.

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Chapter 13.4, Problem 30E

Chapter 13.5, Problem 2E

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