Torque & Equilibrium Torque & Equilibrium Rotation in daily life is everywhere: doors turning on hinges, spanners loosening nuts, and see-saws balancing children. A push does not always translate into motion; it may try to twist. Torque measures this twisting influence of a force about a chosen pivot or axis. The same force can cause large or small rotation depending on where and in what direction it acts. Equilibrium is the condition where a body has no linear acceleration and no angular acceleration. For static equilibrium, nothing moves or turns: the net force is zero and the net torque about any point is zero. These two simple ideas power a huge range of tasks, from designing bridges and ladders that do not collapse, to understanding how muscles create rotation at joints. In this lesson you will connect clear physical pictures (lever arm, pivot, couple) with vector equations, learn the proper sign conventions, and master the classic NEET strategies: choose a smart axis, keep track of perpendicular distance, and eliminate unknown reaction forces efficiently. By the end, you will be able to set up and solve equilibrium problems confidently, and you will know exactly when and how torque creates rotation—and when it cannot. Big-picture feel: A long spanner “magnifies” your effort because it increases the lever arm. Pulling perpendicular to the spanner gives maximum turning effect; pulling along it gives almost none. remember Torque depends on three things: the force, the position vector of the point of application relative to the pivot, and the angle between them. A convenient way to visualize it is the lever arm: the shortest (perpendicular) distance from the pivot to the line of action of the force. The larger this perpendicular distance, the larger the torque for the same force. The direction of torque follows the right-hand rule relative to the axis: in a planar problem, we take one sense (say counterclockwise) as positive. Picking and sticking to one sign convention throughout a problem removes many errors. The rotational effect of a force about a pivot or axis; vector defined by the cross product of position and force. Torque (Moment of force) Lever arm (Moment arm) The perpendicular distance from the pivot/axis to the line of action of the force; = F lever arm . Pivot / Axis Chosen reference point or line about which torques are calculated. Choice is arbitrary but affects algebraic simplicity. A rule to assign positive/negative to torque (e.g., counterclockwise + , clockwise - ) in planar problems. Sign convention In planar (2D) problems about a fixed axis perpendicular to the plane, torque signs are assigned by rotation sense: choose counterclockwise as positive by default unless the question specifies otherwise. For full 3D problems, torque is a vector given by a cross product and its direction is along the axis following the right-hand rule. 5.1 Vector torque Torque is the cross product of the position vector (from pivot to point of application) and the force. Magnitude and lever arm 5.2 Here is the angle between r and F , and r is the perpendicular distance from pivot to the line of action. Two equivalent methods give the same torque magnitude: either multiply r F , or resolve the force into perpendicular and parallel components to r and retain only the perpendicular one. This flexibility is useful when reading geometric data from diagrams. Always draw a quick sketch marking the line of action and the lever arm. Couple Two equal, opposite, and parallel forces separated by a distance. Net force is zero, but there is a non-zero net torque (pure rotation tendency). A couple produces a pure rotational effect without any net force. Its torque magnitude equals the force times the separation (the couple arm). Importantly, the torque of a couple is independent of the choice of origin. This property makes couples handy surrogates for representing engines, steering wheels, or any symmetric twisting action. couple = d F ( independent of origin ) Torque of a couple is independent of origin: couple = d F Two forces are equal in magnitude, opposite in direction, and parallel. Forces act at points whose position vectors differ by d . Couple torque magnitude 5.3 Here d is the perpendicular separation between the two parallel forces. Use this to find the net resultant force acting on a point, which must be zero for the body to be in translational equilibrium. tip A couple’s torque does not change if you shift the origin or the pivot. You can slide a pure couple anywhere on the body without changing its rotational effect. Static equilibrium A body at rest with zero linear acceleration and zero angular acceleration: F = 0 and = 0 about any point. For static equilibrium, both force and torque balances must hold simultaneously. The net force balance controls translational effects; the net torque balance controls rotational effects. In planar problems you typically write two force equations (horizontal and vertical) and one torque equation. The pivot can be chosen freely; a clever choice can eliminate unknown reactions and make the algebra short. 5.4 Equilibrium conditions Zero net force and zero net torque are both required for static equilibrium. This condition holds true when a system is in static or dynamic equilibrium, meaning there is no change in linear or angular velocity. neet-alert Smart pivot choice: Take moments about a point that passes through unknown forces (e.g., a hinge). Their lines of action then have zero lever arm, so they drop out of the torque equation. Draw a clean free-body diagram (FBD). Mark all forces with correct lines of action. Show the center of mass for weight. Choose an axis (or point) for torque balance that eliminates as many unknowns as possible. Write component-wise force balance (e.g., F x = 0 , F y = 0 ). Write torque balance about the chosen point with a clear sign convention. Solve systematically. If friction is involved, use f N and at limiting case f = N . Strategy to solve equilibrium questions Free-body diagrams are crucial. For extended bodies, the weight acts at the center of mass. If strings or rods support the body, their tensions act along their own directions. For a hinge, model unknown reactions by two perpendicular components (and sometimes a moment, if the hinge can exert a couple). For smooth (frictionless) contact surfaces, the normal reaction is perpendicular to the surface and there is no friction. 5.5 Rotational dynamics link About a fixed axis for a rigid body, net external torque equals moment of inertia times angular acceleration. Static equilibrium is the special case of rotational dynamics with = 0 and linear acceleration a = 0 . Then = 0 and F = 0 . remember Angular displacement with constant angular acceleration: = 0 + 0 t + 1 2 t 2 (valid for a fixed axis and constant ; use radians). = 0 + 0 t + 1 2 t 2 = 0 + 0 t + 1 2 t 2 for constant Rigid body rotating about a fixed axis. Angular acceleration is constant. Angles measured in radians. Use radians for angular kinematics; degrees break the calculus link. The formula assumes a fixed axis and constant angular acceleration. If varies with time, integrate (t) appropriately to obtain (t) and (t) . rad Angle theta (rad) Torque for a force applied at fixed r and F varies as tau = r F sin(theta); zero at 0 and pi, maximum at pi/2. custom Torque vs angle for fixed r and F: a sine curve peaking at 90°. Torque (N m) N m Zero torque (force along r) Maximum torque rF pi/2 pi Zero torque (opposite along r) theta control dependent tau N m r = 0.75 m F = 20 N Angle between r and F : = 60 Use = r F and right-hand rule for sign. easy Torque magnitude and sign (counterclockwise positive). A 0.75 m long door is hinged at one edge. A person pulls with a force of 20 N at the outer edge, making a 60° angle with the radius vector from hinge to the handle. Find the torque magnitude about the hinge and its sense. This example shows how the same force creates different torques depending on the angle. Pulling perpendicular to the door radius gives the greatest torque; pulling along the radius gives zero torque. Always check the angle between r and F , not between the force and the surface. Tension T in string; vertical reaction R y at hinge. A uniform horizontal rod of length 2.0 m and mass 8.0 kg is hinged at the left end and held horizontal by a vertical string attached to its right end. Find the tension in the string and the vertical reaction at the hinge. Take g = 9.8 m/s 2 . Static equilibrium: F x=0 , F y=0 , and hinge =0 . Torque about hinge eliminates hinge reactions. medium Rod length L = 2.0 m Mass m = 8.0 kg (weight mg = 78.4 N) String at right end is vertical Hinge at left end Choosing the hinge as the torque point eliminated both unknown hinge reactions instantly. That one line saved two unknowns. This is the standard approach in support-and-hinge problems: take moments about the hinge or any point passing through multiple unknown forces. Minimum angle ( ) such that static equilibrium is possible; numerical value for ( = 0.40 ). A uniform ladder of length L rests against a smooth vertical wall and a rough horizontal floor. The ladder makes an angle ( ) with the floor. If the coefficient of static friction at the floor is ( ) and the wall is smooth (no friction), find the minimum angle ( ) for which the ladder does not slip. Then compute ( ) for ( = 0.40 ). At impending slip, friction at the floor is limiting: (f = N f ). Use F x=0 , F y=0 , and about the floor contact. hard Wall is smooth: friction at wall = 0 (only normal N w , horizontal). Floor is rough: normal N f (vertical) and friction f (horizontal). Weight W acts at the center (L/2 from either end) vertically downward. degrees In limiting equilibrium, replace f N with f = N . The smooth wall guarantees the wall force has no vertical component, greatly simplifying the torque balance. The classic trap is mixing up and in the moment arms; always project distances perpendicular to the forces’ lines of action. Torque of any force is the same about all origins. Torque of a single force depends on the origin because the position vector changes. Only a pure couple has origin-independent torque. If net force is zero, the body cannot rotate. Zero net force does not guarantee zero net torque. A couple produces rotation tendency with net force zero. Both F = 0 and = 0 are required for static equilibrium. Perpendicular distance trap: = F r uses the shortest distance from the pivot to the line of action. Do not plug the raw distance between pivot and the point of application unless it is perpendicular. neet-alert Two-Zero Test for static equilibrium: Zero net Force + Zero net Torque = Static equilibrium. Recall as “2Z: Force Zero, Torque Zero.” Do choose a pivot that cancels multiple unknowns. Do mark lever arms clearly on your diagram. Do keep a consistent torque sign convention. Don’t forget weight acts at the center of mass for a uniform body. Don’t mix up angle with the surface and angle between r and F . Do and Don’t Gravitational torque for a uniform rod of length L about one end equals mg ,(L/2) , where is the angle between the rod and the horizontal. For any extended body, attach the weight at the center of mass when computing torque about any chosen pivot. This simple replacement is an exact result for rigid bodies under uniform gravity. Contrast torque and work despite same unit dimensions. Quantity Definition SI unit Vector? Depends on origin? Force Push/pull causing linear acceleration Yes No Torque (moment of force) Rotational effect: ( = r F ) N m Yes Yes (single force) Couple Two equal opposite parallel forces separated by distance N m Yes (free vector) No Work Energy transfer by force along displacement J (N m) No (scalar) No Although torque and work share the unit N m, their physical meanings differ completely: torque is a pseudovector causing rotation, while work is a scalar energy transfer. Keep their roles distinct when interpreting results. FBD of hinged rod with forces labeled: weight at center, vertical tension at free end, hinge reactions at pivot. Free-body diagram of a horizontal hinged rod with a vertical string at the end; show hinge reactions, tension, and weight at center. Hinged rod in static equilibrium: pick the hinge as pivot to eliminate hinge reactions in the torque equation. Use the visualizer to explore how torque changes with angle and distance. Keep the force fixed and slide the point of application outward: the lever arm grows, and so does the torque. Rotate the force direction: torque follows the sine of the angle, peaking at 90°. Torque of rod’s weight about hinge 5.6 For a uniform rod inclined at angle to the horizontal, weight acts at its midpoint. Stability near equilibrium can be assessed by the sign of the slope of net torque versus small angular displacement. In stable equilibrium, a small displacement generates a restoring torque that tends to bring the body back. Equivalently, stable equilibrium corresponds to a local minimum of potential energy; neutral to a flat region; unstable to a local maximum. Unit trap: Torque has unit N m, same as joule, but it is not energy. Do not report torque answers in joules; write N m and interpret it as a rotational tendency, not work. neet-alert Angular kinematics derived from calculus require radians. Put , , and in radian-based formulas; convert degrees to radians first. You can use degrees freely in rotational kinematics formulas. Recap: Torque is r F , with magnitude rF or Fr . A couple is a pure torque free of net force and independent of origin. Static equilibrium needs F = 0 and = 0 about any point. For speed on NEET, draw an FBD, pick a smart pivot, and write the minimum set of independent equations that kill the unknowns quickly. Key terms Torque Rotational effect of a force about a pivot: = r F . Lever arm Perpendicular distance from pivot to force’s line of action. Couple couple Two equal and opposite parallel forces producing a pure torque. Static equilibrium F = 0 and = 0 ; no linear or angular acceleration. Rotational inertia about an axis; links torque to angular acceleration: = I . Moment of inertia Curl fingers from r to F ; thumb points along direction. Right-hand rule