A closed circular wire hung on a nail in a wall undergoes small oscillations of amplitude `2^@` and time period 2s. Find a the radius of the circular wire. b. the speed of the particle farthest away from the point of suspension as it goes though its mean position c. the aceleration of this particle as ilt goes through its mean position and extreme position. Take `g=pi^2 sms^-2`

Explanation

(a)
`tau=-mgrsintheta`
`I=mr^2+mr^2=2mr^2`
`tau=-mgrtheta` (`:' sin theta=theta`)
`Ialpha=-mgrtheta`
`alpha=-(mgr)/(I)theta=-omega^2theta`
`T=2pisqrt((I)/(mgr))=2pisqrt((2mr^2)/(mgr))=2pisqrt((2r)/(g))`
`T^2=(4pi^2*2r)/(g)`
`r=(gT^2)/(8pi^2)=(10xx(2)^2)/(8xx10)=0.5m=50cm`
(b)
`theta_0=2^@=(2pi)/(180)=(pi)/(90)rad`
`omega=(2pi)/(T)=(2pi)/(2)=pi`
`omega_(max)^'=theta_0omega=(pi)/(90)xxpi=(pi^2)/(90)=(10)/(90)=1/9 rad//s`
`v_(max)=2romega_(max)^'=2xx0.5xx1/9=1/9m//s`
(c) `a_c=(v_(max)^2)/(2r)=((1//9)^2)/(2xx0.5)=(1)/(81)m//s^2`, towards O
(d) In extreme position, `a_c=0`
`a_1=2ralpha=2romega^2theta_0`
`=2xx0.5xx(pi)^2xx(pi)/(90)=pi/9m//s^2`