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From MyMCAT

Contents 1 Introduction 2 Torque 3 Static Equilibrium 4 Example 1 5 Example 2 6 Example 3 7 Example 4 8 Centers of Gravity

Introduction

Just as force is a vector that measures the tendency of an object to accelerate in the direction of the overall net force, torque measures the tendency of a force to rotate an object about an axis. While force can be thought of as a push or a pull on an object which causes the object to move forwards or backwards, torque can be thought of as a push or a pull on an object which causes the object to twist clockwise or counterclockwise.

Torque

The SI unit for torque is newton meters (N m). To determine the torque of a force acting on an object, one needs to identify the magnitude of the force, the distance away from the pivot the force is acting at, and the angle θ between to the force and the distance.

The sign of torque is meaningless in most cases as generally one is simply looking for the magnitude, however by convention, if the sign is positive, it is said to be towards the counterclockwise direction, while negative is in the clockwise direction. It is however extremely important to keep track of signs when multiple torques are affecting an object as one will have to add and subtract them based on their directions.

Static Equilibrium

Just as an object will not accelerate if all the forces acting on it cancel (the net force equals zero) if all the torques sum to zero then the object is said to be in static equilibrium and will not accelerate clockwise or counterclockwise. The object may still be rotating, but its period (or frequency) will be constant as there is no net force twisting the object to accelerate or decelerate it.

Example 1

Consider a block on one side of a lever. It is a distance x away from the pivot point (where the torque will be) and it has a mass of m, thus a force of mg down. The torque then, will simply be the force times the distance. (Since they are perpendicular to each other we can ignore the sinθ.) T = mgx.

Example 2

Consider the same system when the lever is straight up and the mass is resting right on the edge. The distance is still x, but now the force, mg, is in the same direction as x. If we look at the angle, θ, we know it is is zero, thus T = mgxsin(0) = 0. So while a force is acting on it, the force is NOT actually causing it to rotate and thus the torque is zero. (If you are unsure of this, try pushing an open door at its edge such that you are applying force towards the hinges.)

Example 3

At any other position, we must take into account the angle. Thus if the lever is at an angle of θ, gravity is not entirely in the direction of the movement, thus we use the sinθ term to determine the component which is applicable to the torque and we get T = Fdsinθ = mgxsinθ.

Example 4

Now let us look at what happens when two blocks are on the lever. Suppose this system is not rotating. If this is the case, then the net torque must be zero. But we know there are two torques being applied (because there are TWO masses), thus they must be equal and opposite. If we say that counterclockwise acting rotation is positive and clockwise acting rotation is negative, then m1 is acting in a positive direction and m2 is acting in a negative direction. Thus T1 (for m1) - T2 (for m2) = 0. Or F₁x₁sinθ - F₂x₂sinθ = 0, or F₁x₁ = F₂x₂, therefore m₁gx₁ = m₂gx₂.

In general, if there are multiple blocks on the lever and it is not rotating, then we can say that the sum of all the torques acting in the clockwise direction will equal the sum of all the torques acting in the counter clockwise direction.

Centers of Gravity

A final note, for all objects causing a torque, the distance between it and the pivot point always comes from the center of mass. Thus if the object is not a perfect sphere or block (where we just assume it comes form the center) we must actually determine the center mass is.

In some cases, the lever itself has mass. In this scenario, we just treat the plank like any other block and determine the torque originating from its center of mass (the center of the lever). Obviously, if the levers center rests perfectly on the pivot, then the distance used in the torque is zero and we can ignore it, but otherwise it will contribute to the torque just as any other block resting on it would.