MCAT Content / Equilibrium / Torques Lever Arms

Torques, Lever arms

Topic: Equilibrium

The second condition of static equilibrium says that the net torque acting on the object must be zero. An object in static equilibrium is one that has no acceleration in any direction. While there might be motion, such motion is constant. If a given object is in static equilibrium, both the net force and the net torque on the object must be zero.

Torque is the force that causes objects to turn or rotate (i.e., the tendency of a force to rotate an object about an axis). The farther the force is applied from the axis of rotation, the greater the angular acceleration; the effectiveness depends on the angle at which the force is applied; and the magnitude of the force must also be part of the equation. For rotation in a plane, torque has two possible directions: clockwise or counterclockwise relative to the chosen pivot point.

This tendency is measured in general about a point, and is termed as moment of force. The torque in angular motion corresponds to force in translation. It is the “cause” whose effect is either angular acceleration or angular deceleration of a particle in general motion. Quantitatively, it is defined as a vector given by:

Rotation is a special case of angular motion. In the case of rotation, torque is defined with respect to an axis such that vector “r” is constrained as perpendicular to the axis of rotation. In other words, the plane of motion is perpendicular to the axis of rotation. Clearly, the torque in rotation corresponds to force in translation.

Torque is the cross product of force cross length of the lever arm; it is involved whenever there is a rotating object. Torque can also be expressed in terms of the angular acceleration of the object.

The determination of torque’s direction is relatively easier than that of angular velocity. The reason for this is simple: the torque itself is equal to vector product of two vectors, unlike angular velocity which is one of the two operands of the vector product. Clearly, if we know the directions of two operands here, the direction of torque can easily be interpreted.

Since torque depends on both the force and the distance from the axis of rotation, the SI units of torque are newton-meters.

For static equilibrium, the net force acting on the object must be zero. Therefore, all forces balance in each direction. Mathematically, this is stated as Fnet = ma = 0. The second condition necessary to achieve equilibrium involves avoiding accelerated rotation (maintaining a constant angular velocity). A rotating body or system can be in equilibrium if its rate of rotation is constant and remains unchanged by the forces acting on it. n equation form, the magnitude of torque is defined to be τ=rFsinθ where τ (the Greek letter tau) is the symbol for torque, r is the distance from the pivot point to the point where the force is applied, F is the magnitude of the force, and θ is the angle between the force and the vector directed from the point of application to the pivot point.

In equilibrium, the net force and torque in any particular direction equal zero. An object with constant velocity has zero acceleration. A motionless object still has constant (zero) velocity, so motionless objects also have zero acceleration. Newton’s second law states that: F = ma. Objects with constant velocity also have zero net external force. This means that all the forces acting on the object are balanced — that is to say, they are in equilibrium. In any system, unless the applied forces cancel each other out (i.e., the resultant force is zero), there will be acceleration in the direction of the resultant force. In static systems, in which motion does not occur, the sum of the forces in all directions always equals zero.

This rule also applies to rotational motion. If the resultant moment about a particular axis is zero, the object will have no rotational acceleration about the axis. If the object is not spinning, it will not start to spin. If the object is spinning, it will continue to spin at the same constant angular velocity.


Key Points

• The second condition necessary to achieve equilibrium involves avoiding accelerated rotation.

• A rotating body or system can be in equilibrium if its rate of rotation is constant and remains unchanged by the forces acting on it.

• The magnitude of torque about an axis of rotation is defined to be τ=rFsinθ.

• Torque is found by multiplying the applied force by the distance to the axis of rotation, called the moment arm.

• Torque is to rotation as force is to motion.

• The unit of torque is the newton-meter.a

• In equilibrium, the net force in all directions is zero.

• If the net moment of inertia about an axis is zero, the object will have no rotational acceleration about the axis.

• In each direction, the net force takes the form F=ma=0 and the net torque takes the form Στ= Iα=0 where the sum represents the vector sum of all forces and torques acting.


Key Terms

Second condition of static equilibrium: The net torque acting on the object must be zero.

Torque: A rotational or twisting effect of a force; (SI unit newton-meter or Nm; imperial unit foot-pound or ft-lb)

Moment of force: The turning effect of a force applied to a rotational system at a distance from the axis of rotation.

Lever arm: The perpendicular distance from the axis of rotation to the line of action of the force.

Equilibrium: The state of a body at rest or in uniform motion, the resultant of all forces on which is zero.

Vector: A directed quantity, one with both magnitude and direction; the between two points.

Angular velocity: A vector quantity describing an object in circular motion; its magnitude is equal to the speed of the particle and the direction is perpendicular to the plane of its circular motion.

Angular motion: The motion of a body about a fixed point or fixed axis (as of a planet or pendulum). It is equal to the angle passed over at the point or axis by a line drawn to the body.



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