Let the rod be deviated through an angle φ'from its initial position at an arbitrary instant of

Let the rod be deviated through an angle φ'from its initial...

Let the rod be deviated through an angle $\varphi$ 'from its initial position at an arbitrary instant of time, measured relative to the initial position in the positive direction. From the equation of the increment of the mechanical energy of the system. or, $$ \begin{gathered} \Delta T=A_{e x t} \\ \frac{1}{2} I \omega^{2}=\int N_{z} d \varphi \end{gathered} $$ or, $$ \frac{1}{2} \frac{M l^{2}}{3} \omega^{2}=\int_{n}^{\varphi} F l \cos \varphi d \varphi=F l \sin \varphi $$ Thus, $$ \omega=\sqrt{\frac{6 F \sin \varphi}{M l}} $$

Let the rod be deviated through an angle $\varphi$ 'from its initial position at an arbitrary instant of time, measured relative to the initial position in the positive direction. From the equation of the increment of the mechanical energy of the system.
or,
$$
\begin{gathered}
\Delta T=A_{e x t} \\
\frac{1}{2} I \omega^{2}=\int N_{z} d \varphi
\end{gathered}
$$
or,
$$
\frac{1}{2} \frac{M l^{2}}{3} \omega^{2}=\int_{n}^{\varphi} F l \cos \varphi d \varphi=F l \sin \varphi
$$
Thus,
$$
\omega=\sqrt{\frac{6 F \sin \varphi}{M l}}
$$

Key Concepts

Mechanical Energy

Mechanical energy in a rotational system is comprised of rotational kinetic energy and potential energy. In the context of this problem, the increase in mechanical energy is analyzed by comparing the work done by a torque to the change in the rotational kinetic energy, illustrating the conversion of work into kinetic energy.

Moment of Inertia

The moment of inertia quantifies the distribution of mass in a rotating object relative to the axis of rotation. It determines how resistant the object is to changes in its angular motion. In this case, the rod's moment of inertia is used to calculate its rotational kinetic energy, playing a crucial role in connecting the applied torque to the resulting angular speed.

Torque

Torque is the measure of a force's ability to produce rotational acceleration about an axis, defined as the product of force and lever arm length. The work done by a torque is obtained by integrating it over the angular displacement, which directly links the applied force to the change in the system’s rotational energy.

Work-Energy Principle in Rotation

The work-energy principle in rotational dynamics states that the work done by net torque on a rotating body is equal to its change in rotational kinetic energy. This principle is used to derive relationships between the applied force, angular displacement, and the resulting angular speed.

Angular Kinematics

Angular kinematics deals with the description of rotational motion, including angular displacement, angular velocity, and angular acceleration. Understanding these aspects is essential for linking the applied torque and the resulting angular speed, as demonstrated in the derivation of the expression for angular velocity.

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