Torque from Distributed Forces
When force is spread over a line, area, or volume, the torque about an origin is the integral of r cross dF; a single resultant force may replace the distribution only when its line of action is also preserved.
Why this shows up in the exam
Frictional torque on rotating mops and discs · Torque from fluid pressure on surfaces · Replacing a distributed load by a resultant and moment
Learn the idea
A distributed contact produces total torque by summing each small force times its own lever arm. A distributed contact produces total torque by summing each small force times its own lever arm. Start from a clear axis, origin, body, and reference frame; the geometry and constraints then decide which rotational law is safe to use.
🧠 Memory hook: Integrate every force ring with its own radius.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- tau_O = integral r cross dF — Total torque of a continuous force distribution about O.
- dF = p dA or mu p dA — Normal or frictional force element for a pressure distribution.
How to approach it
- 1Choose a differential element suited to the symmetry
- 2Express dF and its perpendicular lever arm
- 3Integrate over the loaded region and check torque units
Common slip-ups that cost marks
- •Applying the total force at the outer radius
- •Averaging radius without force weighting
- •Replacing a distribution by a resultant but losing its moment
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
Masses 1 kg and 3 kg lie at x = 0 and x = 4 m. Find the x-coordinate of their centre of mass.
More from Motion of System of Particles and Rigid Body
Conservation of momentum and angular momentum
The total linear and angular momentum of a system remains constant in the absence of external forces or torques, including during collisions and rotational motion.
Moment of inertia and radius of gyration
Moment of inertia quantifies how mass is distributed with respect to an axis of rotation, and the radius of gyration is a measure related to this distribution.
Torque and rotational equilibrium
Torque is the rotational analogue of force, causing angular acceleration, and equilibrium occurs when the net torque on a body is zero.
Center of mass: definition and calculation
The center of mass is the point representing the mean position of the mass in a system, and can be calculated for discrete particles or continuous bodies.
Rotational kinematics and dynamics
Rotational kinematics describes the motion of rotating bodies, while dynamics relates torque, angular acceleration, and rotational kinetic energy.
Centre of Mass of Discrete Particles
For discrete particles, the centre-of-mass position is the vector sum of each mass times its position divided by total mass; this point governs translation even when the particles move relative to one another.