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  4. A uniform circular ring of mass m and radius R is placed freely on a horizontal smooth surface as shown in figure. A particle of mass m is connected to the circumference of the ring with a massless string. The particle is imparted velocity v0 perpendicular to length of string as shown. If T is tension in the string just after the particle imparted velocity, then <img class=img-fluid question-image alt=Question src=https://cdn.tardigrade.in/q/nta/p-asdo2rhoqcwd.png /> Acceleration of point P at this instant, is

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Q. A uniform circular ring of mass $m$ and radius $R$ is placed freely on a horizontal smooth surface as shown in figure. A particle of mass $m$ is connected to the circumference of the ring with a massless string. The particle is imparted velocity $v_{0}$ perpendicular to length of string as shown. If $T$ is tension in the string just after the particle imparted velocity, then

Question

Acceleration of point $P$ at this instant, is

NTA AbhyasNTA Abhyas 2020System of Particles and Rotational Motion

Solution:

Calculate acceleration of each in ground frame.
$ \, T=ma_{c}$
Solution
$T.R=mR^{2}\alpha $
$T=mR\alpha \, $
$a_{P}=\left(\right.a_{C}+\alpha R\left.\right)$
$T=ma_{c}\Rightarrow a_{c}=\alpha R$
$a_{p}=\frac{2 T}{m}$