A couple is a set of two forces of equal magnitude and opposite directions, whose lines of action are parallel but not identical. Prove that the net torque of a couple of forces is independent of the pivot point about which the torque is calculated and of the points along their lines of action where the two forces are applied.

A couple is defined as a set of two forces of equal magnitude and opposite directions, whose lines of action are parallel but not identical. Torque, also known as the moment of force, measures the tendency of a force to rotate an object around a pivot point. The torque τ is calculated using the formula τ = r × F, where r is the perpendicular distance from the pivot point to the line of action of the force, and F is the force.

Let the couple of forces be F and -F, both parallel to the x-axis but originating at different points along the y-axis. Let the forces be applied at points A and B with coordinates (x1, y1) and (x2, y2), respectively. Let O be the pivot point with coordinates (x0, y0).

To calculate the torque of each force, we need to determine the perpendicular (shortest) distance from the pivot point to the line of action of each force. Let d1 be the perpendicular distance from O to the line of action of force F, and d2 be the perpendicular distance from O to the line of action of force -F. Using the formula for torque, the torque τ1 due to force F is given by τ1 = F * d1, and the torque τ2 due to force -F is given by τ2 = -F * d2.

The net torque of the couple is the sum of the individual torques, i.e., τ_net = τ1 + τ2. Substituting the expressions for τ1 and τ2, we get: τ_net = (F * d1) + (-F * d2) We can factor out the force F from the expression: τ_net = F * (d1 - d2)

To prove that the net torque is independent of the pivot point, we need to show that the value of (d1 - d2) remains constant for any value of (x0, y0). The distance formula can be used to find the perpendicular distance between two lines, as follows: If y = mx + c1 and y = mx + c2 are two parallel lines, then the perpendicular distance between them is given by: d = |c1 - c2| / sqrt(1 + m^2) Since both forces, F and -F, are parallel to the x-axis (i.e., have a slope of 0), we can write their equations as y = 0 * x + y1 (force F) and y = 0 * x + y2 (force -F). Applying the distance formula, we get: d1 - d2 = |y1 - y0| - |y2 - y0| Since the two forces are parallel, (y1 - y0) and (y2 - y0) have the same sign. Therefore, the magnitudes of the distances remain the same, regardless of the pivot point (x0, y0). Thus, we can conclude that the net torque of a couple is independent of the pivot point. To prove that the net torque is independent of the points of application, we can consider how changing the points A and B affects the net torque. Since both forces, F and -F, are parallel, moving A and B along their lines of action does not change their slopes or the distances between them. This means that the net torque remains unchanged.