Question

Physic: A yo-yo of total mass m consists of two solid cylinders of radius R, connected by a small spindle of negligible mass and radius r. The top of the string is held motionless while the string unrolls from the spindle freely under gravity. Given the angular momentum of a cylinder L = πmR2/T, find the linear acceleration of the yo-yo. 8. A yo-yo of total massmconsists of two solid cylinders of radiusR, connected by a small spindle of negligible mass and radiusr. The top of the string is held motionless while the string unrolls from the spindle freely under gravity. Given the angular momentum of a cylinderL=πmR 2/T, find the linear acceleration of the yo-yo.

Answer 1

The moment of inertia can be given as the sum of the moments of inertia of each of the end cylinders plus the moment of inertia of the axle is${a_y = \frac{g}{1+\frac{R^2}{2r^2}}}$

Imagine the YoYo When yoyo is falling

In Linear downward direction,

mg-T=ma_y

ay = is linear acceleration

now

$T*r=I_c_m \alpha$

but from we also know $\alpha =\frac{a_y}{r}$

$T*r=I_c_m \frac{a_y}{r}$

substiting above equation in beggining eqyation.

$mg-I_c_m \frac{a_y}{r^2} = ma_y$

$a_y = \frac{mg}{m+\frac{I_c_m}{r^2}} .......................(1)$

Icm --> The moment of inertia can be given as the sum of the moments of inertia of each of the end cylinders plus the moment of inertia of the axle:

and mass ofcylinders will half of total mass as spindle mass is negigible. i.e cylinders masses = m/2

$I_c_m =2* \frac{\frac{m}{2}R^2}{2} = \frac{mR^2}{2}...............(2)$

Substituting 2 in 1 we get

${ a_y = \frac{g}{1+\frac{R^2}{2r^2}}}$

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