Let us use the equation (d Mz)/(d t)=Nz relative to the axis through O For this purpose, let us

step 1 In this problem, let us use the equation D m z by d t is equal to n z relative to its axis throu

Let us use the equation (d Mz)/(d t)=Nz relative to the axis...

Let us use the equation $\frac{d M_{z}}{d t}=N_{z}$ relative to the axis through $O$ For this purpose, let us find the angular momentum of the system $M_{z}$ about the given rotation axis and the corresponding torque $N_{z}$. The angular momentum is $$ M_{z}=I \omega+m v R=\left(\frac{m_{0}}{2}+m\right) R^{2} \omega $$ [where $I=\frac{m_{0}}{2} R^{2}$ and $v=\omega R$ (no cord slipping)] So, $$ \frac{d M_{z}}{d t}=\left(\frac{M R^{2}}{2}+m R^{2}\right) \beta_{z} $$ The downward pull of gravity on the overhanging part is the only external force, which exerts a torque about the $z$ -axis, passing through $O$ and is given by, $$ N_{z}=\left(\frac{m}{l}\right) x g R $$ Hence from the equation $\quad \frac{d M_{z}}{d t}=N_{z}$ $$ \left(\frac{M R^{2}}{2}+m R^{2}\right) \beta_{z}=\frac{m}{l} x g R $$ Thus, $$ \beta_{z}=\frac{2 m g x}{l R(M+2 m)}>0 $$ Note: We may solve this problem using conservation of mechanical energy of the system (cylinder $+$ thread) in the uniform field of gravity.

Let us use the equation $\frac{d M_{z}}{d t}=N_{z}$ relative to the axis through $O$ For this purpose, let us find the angular momentum of the system $M_{z}$ about the given rotation axis and the corresponding torque $N_{z}$. The angular momentum is
$$
M_{z}=I \omega+m v R=\left(\frac{m_{0}}{2}+m\right) R^{2} \omega
$$
[where $I=\frac{m_{0}}{2} R^{2}$ and $v=\omega R$ (no cord slipping)]
So,
$$
\frac{d M_{z}}{d t}=\left(\frac{M R^{2}}{2}+m R^{2}\right) \beta_{z}
$$
The downward pull of gravity on the overhanging part is the only external force, which exerts a torque about the $z$ -axis, passing through $O$ and is given by,
$$
N_{z}=\left(\frac{m}{l}\right) x g R
$$
Hence from the equation $\quad \frac{d M_{z}}{d t}=N_{z}$
$$
\left(\frac{M R^{2}}{2}+m R^{2}\right) \beta_{z}=\frac{m}{l} x g R
$$
Thus,
$$
\beta_{z}=\frac{2 m g x}{l R(M+2 m)}>0
$$
Note: We may solve this problem using conservation of mechanical energy of the system (cylinder $+$ thread) in the uniform field of gravity.

Key Concepts

Angular Momentum

Angular momentum is a measure of the rotational motion of an object about an axis. It takes into account both the distribution of mass relative to the axis (through the moment of inertia) and the angular velocity. Its conservation is a fundamental principle in rotational dynamics, similar to the conservation of linear momentum in translational motion.

Torque

Torque represents the rotational equivalent of force, quantifying the tendency of a force to rotate an object about an axis. It is calculated as the product of the force and the distance from the axis at which the force is applied, and it is directly related to the rate of change of angular momentum in a system.

Moment of Inertia

The moment of inertia is the rotational analogue of mass in linear motion. It measures how the mass of an object is distributed with respect to a chosen axis and determines the object's resistance to angular acceleration. The greater the moment of inertia, the harder it is to change the object's rotational speed.

Rolling Without Slipping

Rolling without slipping is a condition where the point of contact between a rolling object and the surface does not slide. This condition creates a direct relationship between the translational velocity of the center of mass and the angular velocity of the object, typically represented by the equation v = ?R, linking linear and angular kinematics.

Newton's Second Law for Rotation

Newton's Second Law for rotational motion states that the net torque acting on an object is equal to the time rate of change of its angular momentum. This principle is fundamental in analyzing rotational dynamics and is used to relate applied torques to resulting angular accelerations.

Conservation of Mechanical Energy

Conservation of mechanical energy is the principle that in the absence of non-conservative forces, the total mechanical energy (sum of kinetic and potential energies) of a system remains constant. This principle provides an alternative method to analyze systems involving gravity and rotational motion.

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