Torque — Revision Notes

⚡ 30-Second Revision

Definition: — Rotational effect of a force. \\vec{\tau} = \vec{r} \times \vec{F}\.
Magnitude: — \\tau = rF\sin\theta\ or \\tau = F \cdot r_\perp\.
Moment Arm (\$r_\perp\$): — Perpendicular distance from pivot to line of action of force.
Direction: — Right-hand rule (perpendicular to \\vec{r}\ and \\vec{F}\, along axis of rotation).
Units: — Newton-meter (N\\cdot\m).
Rotational 2nd Law: — \\Sigma \vec{\tau} = I\vec{\alpha}\.
Equilibrium: — \\Sigma \vec{\tau} = 0\ (no angular acceleration).
Relation to Angular Momentum: — \\vec{\tau} = \frac{d\vec{L}}{dt}\.

2-Minute Revision

Torque is the turning effect of a force, crucial for understanding rotational motion. It's a vector quantity, defined as the cross product of the position vector (from the axis of rotation to the point of force application) and the force vector: \\vec{\tau} = \vec{r} \times \vec{F}\.

The magnitude is \\tau = rF\sin\theta\, where 'r' is the distance, 'F' is the force, and \\theta\ is the angle between them. The term \r\sin\theta\ is the 'moment arm' or 'lever arm,' representing the perpendicular distance from the pivot to the line of action of the force.

Torque is maximum when the force is perpendicular to the moment arm (\\theta = 90^\circ\) and zero when parallel (\\theta = 0^\circ\ or \\theta = 180^\circ\). Its direction is given by the right-hand rule, pointing along the axis of rotation.

The SI unit is N\\cdot\m. Torque is the cause of angular acceleration, related by \\Sigma \vec{\tau} = I\vec{\alpha}\, where 'I' is the moment of inertia. For rotational equilibrium, the net torque must be zero.

5-Minute Revision

Torque, often called the moment of force, is the rotational analogue of linear force. It's what causes an object to rotate or change its rotational motion. Mathematically, it's a vector quantity defined by the cross product \\vec{\tau} = \vec{r} \times \vec{F}\.

Here, \\vec{r}\ is the position vector from the axis of rotation to the point where the force \\vec{F}\ is applied. The magnitude of torque is given by \\tau = rF\sin\theta\, where 'r' is the magnitude of the position vector, 'F' is the magnitude of the force, and \\theta\ is the angle between \\vec{r}\ and \\vec{F}\.

A key concept is the 'moment arm' (or lever arm), which is the perpendicular distance from the axis of rotation to the line of action of the force, equal to \r\sin\theta\. Torque is maximized when the force is applied perpendicular to the position vector (\\theta = 90^\circ\), and it's zero when the force acts along the line passing through the pivot (\\theta = 0^\circ\ or \\theta = 180^\circ\).

\n\nThe direction of torque is crucial and is determined by the right-hand rule: point your fingers along \\vec{r}\, curl them towards \\vec{F}\, and your thumb points in the direction of \\vec{\tau}\, which is along the axis of rotation.

For 2D problems, counter-clockwise torque is often taken as positive, and clockwise as negative. The SI unit for torque is Newton-meter (N\\cdot\m), distinct from Joules (energy) despite dimensional similarity.

\n\nTorque plays a central role in rotational dynamics through Newton's second law for rotation: \\Sigma \vec{\tau}_{ext} = I\vec{\alpha}\, where \\Sigma \vec{\tau}_{ext}\ is the net external torque, 'I' is the moment of inertia of the object about the axis of rotation, and \\vec{\alpha}\ is the angular acceleration.

This equation allows us to calculate how much an object will angularly accelerate under a given torque. For an object to be in rotational equilibrium (no angular acceleration), the net external torque acting on it must be zero (\\Sigma \vec{\tau}_{ext} = 0\).

This principle is vital for solving problems involving balanced systems like levers, beams, and ladders. Torque is also the rate of change of angular momentum (\\vec{\tau} = \frac{d\vec{L}}{dt}\), making it fundamental to the conservation of angular momentum principle.

Prelims Revision Notes

1
Definition: — Torque (\\vec{\tau}\) is the rotational effect of a force. It's a vector quantity.\n2. Formula (Vector): \\vec{\tau} = \vec{r} \times \vec{F}\. Remember the order: position vector first, then force vector.\n3. Formula (Magnitude): \\tau = rF\sin\theta\. Here, 'r' is the distance from the pivot to the point of force application, 'F' is the force magnitude, and \\theta\ is the angle between \\vec{r}\ and \\vec{F}\. \\sin\theta\ is crucial.\n4. Moment Arm (Lever Arm): \r_\perp = r\sin\theta\. This is the perpendicular distance from the axis of rotation to the line of action of the force. So, \\tau = F \cdot r_\perp\. This is often easier to visualize.\n5. Maximum Torque: Occurs when \\theta = 90^\circ\ (force is perpendicular to \\vec{r}\). \\tau_{max} = rF\.\n6. Zero Torque: Occurs when \\theta = 0^\circ\ or \\theta = 180^\circ\ (force is parallel or anti-parallel to \\vec{r}\, acting through the pivot). \\tau = 0\.\n7. Direction of Torque: Use the Right-Hand Rule for cross products. Point fingers along \\vec{r}\, curl towards \\vec{F}\, thumb gives \\vec{\tau}\ direction (along the axis of rotation). For 2D, CCW is usually positive, CW is negative.\n8. Units: SI unit is Newton-meter (N\\cdot\m). Do not confuse with Joule (energy), though dimensions are similar.\n9. Rotational Analog of Newton's 2nd Law: \\Sigma \vec{\tau}_{ext} = I\vec{\alpha}\. \\Sigma \vec{\tau}_{ext}\ is the net external torque, 'I' is the moment of inertia, \\vec{\alpha}\ is angular acceleration.\n10. Rotational Equilibrium: For an object to be in rotational equilibrium (no angular acceleration), the net external torque must be zero: \\Sigma \vec{\tau}_{ext} = 0\. This means sum of CCW torques = sum of CW torques.\n11. Relation to Angular Momentum: \\vec{\tau}_{net} = \frac{d\vec{L}}{dt}\. If \\vec{\tau}_{net} = 0\, then angular momentum \\vec{L}\ is conserved.\n12. Problem-Solving Tip: For equilibrium problems, choose the pivot strategically (e.g., at an unknown force location) to simplify calculations by eliminating that force from the torque equation. Always draw a clear free-body diagram and identify all forces and their distances from the chosen pivot.

Vyyuha Quick Recall

To remember the factors affecting torque: 'FAT'

Force (magnitude of the force)
Angle (sine of the angle between force and position vector)
Turning Arm (the moment arm or lever arm, which is the perpendicular distance from the pivot)