Question

A thread is wound around two discs on either sides. The pulley and the two discs have the same mass and radius . There is no slipping at the pulley and no friction at the hinge. Find out the accelerations of the two discs and the angular acceleration of the pulley ​

Answer 1

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$\bf\pink{\mid{\overline{\underline{SOLUTION:-}}}\mid}$

→ Let R be the radius of the discs and T1 and T2 be the tensions in the left and right segments of the rope.

♦ Acceleration of disc 1 ,

$\implies a1 = \frac{mg -T1}{m} ..(1)$

♦ Acceleration of disc 2 ,

$\implies a2 = \frac{mg -T2}{m} ..(2)$

♦ Angular acceleration of disc 1 ,

$\implies a1 = \frac{\pi}{i} = \frac{T1R}{ \frac{1}{2} mR ^{2} } = \frac{2T1}{ mR} ..(3)$

Similarly,

♦ Angular acceleration of disc 2 ,

$\implies a2 = \frac{2T2}{mR} ....(4)$

→ both a1 and a2 are clockwise.

♦ Angular acceleration of pulley

$\implies a= \frac{(T2-T1)R}{ \frac{1}{2}mR^{2} } = \frac{2(T2-T1)}{mR} ..(5)$

For no slipping,

→ Ra1 - a1 = a2 - Ra2 = Ra ...(6)

Solving these equations we get ,

$\bf\implies a = 0$

and

$\bf\implies a1 = a2 = \frac{2 g}{3}$

Answer 2

$\huge \implies \bf{a = 0}$

and

$\bf\huge\implies a1 = a2 = \frac{2 g}{3}$

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