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Kinetic energy of a particle moving alon...

Kinetic energy of a particle moving along a circle of radisu R depends on the distance covered as `K =as^(2)` where a is a constant . Find the force acting on the particle as a function of s.

A

`(2a)/(s)sqrt(1+((s)/(R))^(2))`

B

`2assqrt(1+((R)/(s))^(2))`

C

`2assqrt(1+((s)/(R))^(2))`

D

`(2s)/(a)sqrt(1+((R)/(s))^(2))`

Text Solution

Verified by Experts

(c) According to given problem
`(1)/(2)Mv^(2)=as^(2)` " " [Hence, M =mass]
`rArr" " v=ssqrt((2a)/(M))" " (i)`
So, `" " a_(g) =(v^(2))/(R)=(2as^(2))/(MR)`
Further more as `a_(t)=(dv)/(dt)=(dv)/(ds),(ds)/(dt)=v(dv)/(ds)` (By chain rule)
from Eq. (i) we get
`i,e" " v=sqrt((2a)/(M))` yeilds
`:." " a_(t)=[ssqrt((2a)/(M))][sqrt((2a)/(M))]=(2as)/(M) " " ( :.v/s=sqrt((2a)/(m)))`
so that," " `a_("net")=sqrt(a_(R)^(2)+a_(t)^(2))`
`=sqrt(((2as^(2))/(MR))^(2)+((2a)/(M))^(2))=(2as)/(M)(sqrt(1+((s)/(R))^(2)))`
`:. " " F-Ma_("net")=2assqrt(1+(s//R)^(2))`
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