a-particle-of-mass-m-kg-is-moving-along-a-smooth-wire-that-is-fixed-in-a-plane-The-polar-equation-of-the-wire-is-r-ae-3-The-particle-moves-with-a-cons-tant-velocity-of-6-At-time-t-0-th

Question Number 58171 by pierre last updated on 19/Apr/19

a particle of mass m kg is moving along

a smooth wire that is fixed in a plane .

The polar equation of the wire is

r = {ae}^{3 θ} . The particle moves with a cons

tant velocity of 6 . At time t = 0, the par

ticle is at the point with polar equation

(a, θ)

a) Find the transverse and radial compo

nents of the acceleration of the particle

in terms of a and t .

b) the resultant force on the particle is

F . Show that the magnitude of F at time

t is {360 mae}^{18 t}

Answered by tanmay last updated on 19/Apr/19

a) {\vec{a}}_{r} = \overset{. .}{r} - r {(\overset{. .}{θ})}^{2} \to r a d i a l a c c e l a r a t i o n

{\vec{a}}_{T} = 2 \dot{r} \dot{θ} + r \overset{. .}{θ} \to t r a n s v e r s a c c e l a r a t i o n

n o w \dot{r} = \frac{d r}{d t} = \frac{d}{d t} ({a e}^{3 θ}) = a \times e^{3 θ} \times 3 = 3 {a e}^{3 θ} \times \frac{d θ}{d t}

\dot{r} = 3 {a e}^{3 θ} \times \dot{θ}

\overset{. .}{r} = \frac{d}{d t} (\dot{r}) = \frac{d}{d t} (3 {a e}^{3 θ} \times \dot{θ})

\overset{. .}{r} = 3 a [e^{3 θ} \times \overset{. .}{θ} + e^{3 θ} \times 3 \times \frac{d θ}{d t} \times \dot{θ}]

\overset{. .}{r} = 3 {a e}^{3 θ} \overset{. .}{θ} + 9 {a e}^{3 θ} {(\overset{. .}{θ})}^{2} \to 3 r \overset{. .}{θ} + 9 r {(\overset{. .}{θ})}^{2}

Extra \left or missing \right

{\vec{a}}_{r} = 3 r \overset{. .}{θ} + 9 r {(\overset{. .}{θ})}^{2} - r {(\overset{. .}{θ})}^{2} \to 3 r \overset{. .}{θ} + 8 r {(\overset{. .}{θ})}^{2} \leftarrow r a d i a l a c c e l a r a t i o n

{\vec{a}}_{T} = 2 \dot{r} \dot{θ} + r \overset{. .}{θ}

{\vec{a}}_{T} = 2 (3 {a e}^{3 θ} \dot{θ}) \dot{θ} + ({a e}^{3 θ}) \overset{. .}{θ}

{\vec{a}}_{T} = 6 {a e}^{3 θ} \dot{θ} + {a e}^{3 θ} \overset{. .}{θ}

{\vec{a}}_{T} = 6 r \dot{θ} + \overset{}{r} \overset{. .}{θ} \to t r a n s v e r s e a c c

p l s c h e c k \dots

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