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^(3θ) . 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 360mae^(18t)


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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

	$a) {\vec{a}}_{r} = \overset{. .}{r} - r {(\overset{. .}{θ})}^{2} \to r a d i a l a c c e l a r a t i o n$ ${\vec{a}}_{T} = 2 \dot{r} \dot{θ} + r \overset{. .}{θ} \to t r a n s v e r s a c c e l a r a t i o n$ $n o w \dot{r} = \frac{d r}{d t} = \frac{d}{d t} ({a e}^{3 θ}) = a \times e^{3 θ} \times 3 = 3 {a e}^{3 θ} \times \frac{d θ}{d t}$ $\dot{r} = 3 {a e}^{3 θ} \times \dot{θ}$ $\overset{. .}{r} = \frac{d}{d t} (\dot{r}) = \frac{d}{d t} (3 {a e}^{3 θ} \times \dot{θ})$ $\overset{. .}{r} = 3 a [e^{3 θ} \times \overset{. .}{θ} + e^{3 θ} \times 3 \times \frac{d θ}{d t} \times \dot{θ}]$ $\overset{. .}{r} = 3 {a e}^{3 θ} \overset{. .}{θ} + 9 {a e}^{3 θ} {(\overset{. .}{θ})}^{2} \to 3 r \overset{. .}{θ} + 9 r {(\overset{. .}{θ})}^{2}$ $Extra \left or missing \right$ ${\vec{a}}_{r} = 3 r \overset{. .}{θ} + 9 r {(\overset{. .}{θ})}^{2} - r {(\overset{. .}{θ})}^{2} \to 3 r \overset{. .}{θ} + 8 r {(\overset{. .}{θ})}^{2} \leftarrow r a d i a l a c c e l a r a t i o n$ ${\vec{a}}_{T} = 2 \dot{r} \dot{θ} + r \overset{. .}{θ}$ ${\vec{a}}_{T} = 2 (3 {a e}^{3 θ} \dot{θ}) \dot{θ} + ({a e}^{3 θ}) \overset{. .}{θ}$ ${\vec{a}}_{T} = 6 {a e}^{3 θ} \dot{θ} + {a e}^{3 θ} \overset{. .}{θ}$ ${\vec{a}}_{T} = 6 r \dot{θ} + \overset{}{r} \overset{. .}{θ} \to t r a n s v e r s e a c c$ $p l s c h e c k . . .$
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