Newtonian Mechanics | Lecture: 19 | physics 1st paper chapter 4| Newtonian Mechanics|© Apurbo physics

Torque (Rotational Tendency)
📌 Torque ($\tau$ or $\text{Ta}$) is the rotational equivalent of force ($F$) in linear motion; force causes a change in linear momentum, while torque causes a change in angular momentum ($L$).
📐 The definition of torque is the vector cross product of the radius vector ($r$) and the applied force vector ($F$): $\vec{\tau} = \vec{r} \times \vec{F}$.
📏 The magnitude of torque is given by $\tau = rF \sin\theta$, where $\theta$ is the angle between $r$ and $F$.
💡 Torque is required to change an object's state of rotation—starting it, stopping it, or changing its rotational speed.

Factors Affecting Torque
📌 Torque is directly proportional to the magnitude of the applied force ($F$): greater force results in greater rotational tendency.
📌 Torque is directly proportional to the distance from the axis of rotation ($r$): applying force further from the axis (like the door handle further from the hinge) increases torque.
📌 Torque is directly proportional to $\sin\theta$, where $\theta$ is the angle between $r$ and $F$; maximum torque occurs when $r$ and $F$ are perpendicular ($\theta = 90^\circ$), and torque is zero when they are parallel ($\theta = 0^\circ$ or $180^\circ$).

Couples (Dwandwa/Yugal)
📌 A couple is defined by three conditions: two forces of equal magnitude, parallel to each other, and acting in opposite directions on the body.
📐 The net torque ($\tau_{net}$) produced by a couple is calculated as $\tau_{net} = F \times d$, where $F$ is the magnitude of one force and $d$ is the perpendicular distance between the two parallel forces.
🔄 Both forces in a couple produce torque in the same direction, resulting in a net torque that causes rotation without linear translation.

Angular Momentum ($L$)
📌 Angular momentum ($L$) is the rotational equivalent of linear momentum ($P = mv$); it is defined as the vector cross product of the radius vector ($r$) and the linear momentum ($P$): $\vec{L} = \vec{r} \times \vec{P}$.
💡 An object can possess angular momentum even if it is moving in a straight line, depending on the point of observation (the reference point).
📏 The magnitude of angular momentum is $L = rP \sin\theta = rmv \sin\theta$.

Rotational Dynamics Analogies and Proofs
📌 Linear equations have rotational equivalents:
- $F = ma$ is analogous to $\tau = I\alpha$ (where $I$ is the moment of inertia and $\alpha$ is the angular acceleration).
- $P = mv$ is analogous to $L = I\omega$ (where $\omega$ is the angular velocity).
- Kinetic Energy $K = \frac{1}{2}mv^2$ is analogous to $K = \frac{1}{2}I\omega^2$.
📐 The derivation for $\tau = I\alpha$ was shown by summing the individual torques ($\tau_i = r_i F_i \sin 90^\circ$) for infinitesimal mass elements ($m_i$) of a rotating disc, leading to $\sum \tau_i = (\sum m_i r_i^2) \alpha$.
💡 For a rotating body, only the tangential acceleration ($a_t$) contributes to the torque, as the radial (centripetal) acceleration is directed along the radius vector ($r$), resulting in zero torque ($\sin 0^\circ = 0$).

Key Points & Insights
➡️ For any moving object, its angular momentum ($L$) must be calculated relative to a chosen observation point (reference point).
➡️ The correct vector relationship for torque is $\vec{\tau} = \vec{r} \times \vec{F}$, not $\vec{\tau} = \vec{F} \times \vec{r}$, to ensure the direction aligns with the Right-Hand Rule/physical observation.
➡️ The rotational form of Newton's First Law states: If no net external torque is applied, an object at rest will remain at rest (or an object rotating will continue rotating at a constant angular velocity).
➡️ The assignment requires students to derive the rotational forms of Newton's three laws, specifically proving $\tau = \frac{dL}{dt}$, where $\tau$ is the net external torque.

📸 Video summarized with SummaryTube.com on Jan 23, 2026, 08:54 UTC