Rotational Equilibrium and Rotational Dynamics

Torque

Torque is a vector physical quantity used as a measure of the rotational effect of force. It is proportional to the magnitude of the force and to the perpendicular distance between the point of rotation and the line of action of the force.

|τ| = |F|r⊥

Where |τ| and |F| represent the magnitude of the torque and force respectively; and r⊥ represents the perpendicular distance between the point of rotation and the line of action of the force. If r is the distance between the point of rotation and the point of application of the force and θ is the angle between the position vector of the point of application of force with respect to the point of rotation ( The vector whose tail is at the point of rotation and whose head is at the point of application of force), then r⊥ = r sin θ. Thus

|τ| = |F|r sin θ

The direction of torque is perpendicular to the plane determined by the force vector and the position vector of the point of application of force (with respect to the point of rotation). It is perpendicularly out if the tendency of the force is to produce counterclockwise rotation and perpendicularly in if the tendency of the force is to produce clockwise rotation. The component of torque (τ) is taken to be positive if the tendency of the force is to produce counterclockwise rotation and negative if the tendency of the force is to produce clockwise rotation.

τ = ±|F|r sin θ

The unit of measurement for torque is Newton meter (N m).

Net Torque

Net torque acting on an object is the vector sum of all the torques acting on the object.

τnet = τ1 + τ2 + . . .

If the object is rotating in a plane, all the torques have the same line of action (either perpendicularly out or perpendicularly in) and this vector equation can be described by a single component equation.

τnet = τ1 + τ2 + . . .

Rotational Equilibrium

There are two kinds of equilibriums: translational and rotational equilibrium. An object is said to be in translational equilibrium if it is either at rest or moving in a straight line with a constant speed. The condition of translational equilibrium states that an object will be in translational equilibrium if the net force acting on it is zero.

∑ F = F1 + F2 + . . . = 0

If a vector is equal to zero, then its components are also equal to zero. The following equations are statements of the condition of translational equilibrium in component form.

∑ Fx = F1x + F2x + . . . = 0

∑ Fy = F1y + F2y + . . . = 0

An object is said to be in rotational equilibrium if it is either at rest or rotating with a constant angular speed. The condition of rotational equilibrium states that an object will be in rotational equilibrium if the net torque acting on it is equal to zero.

∑ τ = τ1 + τ2 + . . . = 0

If the rotation is in a plane, the torques will have the same line of action ( perpendicularly out or in ) and this vector equation can be represented by a single ( one dimensional ) component equation.

∑ τ = τ1 + τ2 + . . . = 0

If an object is in equilibrium both translational and rotational conditions of equilibrium apply. For an object in rotational equilibrium, an arbitrary point may be chosen as a point of rotation.

Center of Gravity

The center of gravity of an object is the point at which the object can be balanced. For uniform (constant density) objects, the center of gravity is the same as the geometrical center. For example the center of gravity of a uniform circular disc is at the center of the circle. For problems involving torque, where the point of application of the force has to be specified, the weight of an object can be assumed to act at the center of gravity of the object.

The fact that an object can be balanced at the center of gravity indicates that the torque due to the balancing force (which is equal to the weight) is equal to the sum of the torques of the weights of the particles comprising the object. Taking the torques about the origin of a certain coordinate system, the perpendicular distances are equal to the x-coordinates because the forces are vertical.

M|g|xg = m1|g|x1 + m2|g|x2 + . . .

where M = m1 + m2 + . . ., is the total mass of the object, xi is the x-coordinate of the i th particle and mi is the mass of the i th particle. Thus, the x-coordinate of its center of gravity is given by

xg = ( ∑i mixi ) ⁄ ( ∑i mi ) = ( m1x1 + m2x2 + . . . ) ⁄ ( m1 + m2 + . . . )

A similar expression can be obtained for the y-coordinate of the center of gravity by rotating the object along with the coordinate system by 90°.

yg = ( ∑i miyi ) ⁄ ( ∑i mi ) = ( m1y1 + m2y2 + . . . ) ⁄ ( m1 + m2 + . . . )