Given, M=20 kg ω=100 rad / s R=0.25 m Moment of inertia of the solid cylinder about its ax

step 1 Okay, as per the question, a solid cylinder having mass 20 KG, rotates about it axis with an ang

Given, M=20 kg ω=100 rad / s R=0.25 m Moment of...

Given, $M=20 \mathrm{~kg}$ $$ \begin{gathered} \omega=100 \mathrm{rad} / \mathrm{s} \\ R=0.25 \mathrm{~m} \end{gathered} $$ Moment of inertia of the solid cylinder about its axis of symmetry. $$ \begin{gathered} \quad I=\frac{1}{2} M R^{2} \\ =\frac{1}{2} \times 20 \times(0.25)^{2} \\ =10 \times 0.0625 \\ =0.625 \mathrm{~kg}-\mathrm{m}^{2} \end{gathered} $$ Kinetic energy associated with the rotation of the cylinder is given by $$ K=\frac{1}{2} / \omega^{2} $$ $$ \begin{aligned} &=\frac{1}{2} \times 0.625 \times(100)^{2} \\ &=0.3125 \times 10000 \\ &=3125 \mathrm{~J} \end{aligned} $$ Angular momentum, $L=/ \omega$ $$ \begin{aligned} &=0.625 \times 100 \\ &=62.5 \mathrm{~J}-\mathrm{s} \end{aligned} $$

Given, $M=20 \mathrm{~kg}$
$$
\begin{gathered}
\omega=100 \mathrm{rad} / \mathrm{s} \\
R=0.25 \mathrm{~m}
\end{gathered}
$$
Moment of inertia of the solid cylinder about its axis of symmetry.
$$
\begin{gathered}
\quad I=\frac{1}{2} M R^{2} \\
=\frac{1}{2} \times 20 \times(0.25)^{2} \\
=10 \times 0.0625 \\
=0.625 \mathrm{~kg}-\mathrm{m}^{2}
\end{gathered}
$$
Kinetic energy associated with the rotation of the cylinder is given by
$$
K=\frac{1}{2} / \omega^{2}
$$
$$
\begin{aligned}
&=\frac{1}{2} \times 0.625 \times(100)^{2} \\
&=0.3125 \times 10000 \\
&=3125 \mathrm{~J}
\end{aligned}
$$
Angular momentum, $L=/ \omega$
$$
\begin{aligned}
&=0.625 \times 100 \\
&=62.5 \mathrm{~J}-\mathrm{s}
\end{aligned}
$$

Key Concepts

Angular Momentum

Angular momentum is a measure of the quantity of rotation of an object and is calculated by the product of the moment of inertia and the angular velocity (L = I?) for rigid bodies. It is a conserved quantity in a closed system, meaning that in the absence of external torques, the total angular momentum remains constant.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion about a specific axis. It depends on both the mass of the object and the distribution of that mass relative to the axis of rotation. For standard shapes, such as a solid cylinder, formulas like I = (1/2)MR^2 can be used, which are derived from integrating over the mass distribution.

Angular Velocity

Angular velocity quantifies the rate at which an object rotates or revolves relative to a specified axis. It is usually measured in radians per second. This concept is central to rotational dynamics, as it relates directly to other quantities like kinetic energy and angular momentum in rotating bodies.

Rotational Kinetic Energy

Rotational kinetic energy is the energy due to the rotation of an object and is given by the formula K = (1/2)I?^2, where I is the moment of inertia and ? is the angular velocity. This energy represents the work required to spin up an object from rest up to its current rate of rotation.

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