Question 383: Centre of Mass and Rotational Motion

Question

Moment of inertia about an axis
AB
for a rod of mass 40 kg and length 3 m is same as that of a solid sphere of mass of 10 kg and radius $R$ about an axis parallel to
AB
axis with separation of 3 m as shown in figure below. The value of $R$ is given as $\sqrt{\frac{\alpha}{2}}$. The value of $\alpha$ is ________.

Accepted Answer

Mass of flywheel, $\mathrm{M}=3 \mathrm{~kg}$
Radius of flywheel, $\mathrm{R}=5 \mathrm{~m}$
Descending mass $\mathrm{m}=3 \mathrm{~kg}$
Height through which the mass is descended, $h=3 \mathrm{~m}$
Initial velocity $\mathrm{u}=0 \mathrm{~m} / \mathrm{s}$
Using conservation of energy,
Loss in Potential Energy $=$ Gain in Kinetic Energy of mass + Gain in Rotational Kinetic Energy of wheel
$$ \mathrm{mgh}=\frac{1}{2} \mathrm{mv}^2+\frac{1}{2} \mathrm{I} \omega^2 $$
Since the string does not slip:
$$ \mathrm{v}=\mathrm{R} \omega $$
Moment of Inertia (I) of the flywheel (disk) $=\mathrm{I}=\frac{1}{2} \mathrm{MR}^2$
$$ \mathrm{mgh}=\frac{1}{2} \mathrm{mv}^2+\frac{1}{2}\left(\frac{1}{2} \mathrm{MR}^2ight)\left(\frac{\mathrm{v}}{\mathrm{R}}ight)^2 $$
$$\Rightarrow $$$$ m g h=\frac{1}{2} m v^2+\frac{1}{4} M v^2 $$
Substituting the values :
$$ (3)(10)(3)=\frac{1}{2}(3) v^2+\frac{1}{4}(3) v^2 $$
$$\Rightarrow $$ $90=1.5 \mathrm{v}^2+0.75 \mathrm{v}^2$
$$\Rightarrow $$ $90=2.25 \mathrm{v}^2$
$$\Rightarrow $$ $$ \mathrm{v}^2=\frac{90}{2.25}=40 $$
The kinetic energy of the wheel is the rotational part of the energy :
$\mathrm{K} . \mathrm{E}_{\cdot 	ext { wheel }}=\frac{1}{2} \mathrm{I} \omega^2=\frac{1}{2}\left(\frac{1}{2} \mathrm{MR}^2ight)\left(\frac{\mathrm{V}}{\mathrm{R}}ight)^2=\frac{1}{4} \mathrm{Mv}^2$
$$\Rightarrow $$ K. E.
wheel
$=\frac{1}{4} 	imes 3 	imes 40$
$$\Rightarrow $$ $$ \mathrm{K} . \mathrm{E}_{	ext {.wheel }}=3 	imes 10=30 \mathrm{~J} $$
The kinetic energy of the flywheel when the mass has descended by 3 m is 30 J.
Hence, the correct answer is 30.

Centre of Mass and Rotational Motion

Integer / Numeric Answer Type