Question

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.

Short answer

Question: Prove that the net torque of a couple of forces is independent of the pivot point and the points along their lines of action where the forces are applied. Answer: We have shown that the net torque of a couple of forces can be expressed as \(T_{net} = |\vec{r_1} \times \vec{F_1}| - |\vec{r_2} \times \vec{F_1}|\), which is independent of the pivot point and the points along their lines of action where the forces are applied. Therefore, the net torque of a couple of forces is indeed independent of these factors.

Step-by-step solution

Define torque and couple of forces

Torque (\(T\)) is the turning effect of a force on an object around a particular point known as the pivot point (\(P\)). Torque is mathematically defined as the cross product of the position vector (\(\vec{r}\)) of a point on the line of action of the force and the force vector (\(\vec{F}\)). That is, $$T = |\vec{r} \times \vec{F}|$$ A couple is a set of two forces of equal magnitude and opposite directions, whose lines of action are parallel but not identical. Let \(\vec{F_1}\) and \(\vec{F_2}\) be the two forces in the couple, with magnitudes \(F\) and position vectors \(\vec{r_1}\) and \(\vec{r_2}\) respectively, such that \(\vec{F_2} = -\vec{F_1}\) and \(F = |\vec{F_1}| = |\vec{F_2}|\).

Set up the equation for the net torque

The net torque (\(T_{net}\)) of the couple is the sum of the torque due to the two forces: $$T_{net} = |\vec{r_1} \times \vec{F_1}| + |\vec{r_2} \times \vec{F_2}|$$ Since \(\vec{F_2} = -\vec{F_1}\), we can rewrite the equation for the net torque as: $$T_{net} = |\vec{r_1} \times \vec{F_1}| - |\vec{r_2} \times \vec{F_1}|$$

Use vector algebra to show the net torque is independent of pivot point

Let's represent the position vector from the pivot point (\(P\)) to any point on the line of action of \(\vec{F_1}\) as \(\vec{r_1} + \vec{d}\), where \(\vec{d}\) is an arbitrary vector parallel to \(\vec{F_1}\). We can now use the properties of cross product to rewrite the equation for \(T_{net}\): $$T_{net} = |(\vec{r_1} + \vec{d}) \times \vec{F_1}| - |\vec{r_2} \times \vec{F_1}|$$ $$T_{net} = |\vec{r_1} \times \vec{F_1} + \vec{d} \times \vec{F_1}| - |\vec{r_2} \times \vec{F_1}|$$ Since \(\vec{d}\) is parallel to \(\vec{F_1}\), their cross product is zero: $$\vec{d} \times \vec{F_1} = \vec{0}$$ Thus, the net torque becomes: $$T_{net} = |\vec{r_1} \times \vec{F_1}| - |\vec{r_2} \times \vec{F_1}|$$ Which is independent of the pivot point \(P\) and the points along their lines of action where the forces are applied. In conclusion, we have shown that the net torque of a couple of forces is independent of the pivot point and of the points along their lines of action where the two forces are applied.

Key concepts

Understanding Torque in Physics

In physics, the concept of torque plays a crucial role, especially when discussing the motion of objects under the influence of forces. Torque, often referred to as the moment of force, quantifies how effectively a force can rotate an object around a pivot point. This concept is not only fundamental in mechanics but also has practical applications in our everyday lives, such as turning a doorknob or using a wrench.

The mathematical definition of torque involves the cross product, which interlinks the position vector of the point where the force is applied and the force vector itself. The formula for torque is expressed as \( T = |\vec{r} \times \vec{F}| \) where \( \vec{r} \) is the position vector from the pivot to the point of force application, and \( \vec{F} \) is the force vector. The position vector plays a crucial role, as changing its direction or magnitude can significantly alter the calculated torque. Understanding this relationship helps to grasp how different forces can affect the rotation of an object.

The Cross Product in Physics

The cross product is a vector operation essential in physics, particularly when dealing with rotational motions and torques. Defined only in three dimensions, the cross product combines two vectors to produce a third vector that is perpendicular to both original vectors.

In the context of torque, the cross product is what links the force applied to an object to its rotational effect about a certain point. The result of a cross product gives both the magnitude and the direction of the torque vector. For example, if a force is applied perpendicular to the position vector, the magnitude of the torque is maximized, as the sine of the angle between them, which is a part of the cross product formula, reaches its maximum value of 1.

It's also worth noting that the cross product of two parallel vectors is zero, which comes into play when understanding why a couple's net torque is not affected by its position along their lines of action or the choice of pivot point.

Net Torque Calculation

Net torque calculation often confuses students, but understanding it is vital for resolving complex problems involving rotational motion. The net torque is the vector sum of all individual torques acting on an object. It gives us an overall picture of how an object is going to rotate and about which axis.

To calculate net torque, we sum up the torques produced by each force. If the forces are part of a couple — two equal and opposite forces whose lines of action do not coincide — their individual torques add up in a way that their net effect is independent of the chosen pivot point. This is because the torques they produce around any pivot are equal in magnitude and opposite in direction, only differing in their line of action. Therefore, their individual effects on rotational motion cancel out, leaving a consistent net torque no matter where the pivot point is situated.

Vector Algebra in Physics

Vector algebra is an indispensable tool in physics, as it provides a way to mathematically manipulate quantities that have both magnitude and direction - vectors. In classical mechanics, vectors are used to represent a myriad of physical quantities, including force, velocity, and acceleration.

For problems involving torque, vector algebra is used to determine the direction and magnitude of the rotational effect. One key property of vector multiplication, particularly the cross product, is its distributive nature over addition, which allows us to break down complex torque calculations into simpler parts. Furthermore, it's the aspect of vector algebra that shows the cross product of a parallel force and displacement vector is zero, leading to conclusions about the independent nature of a couple's net torque from the pivot point and the points of force application.

By mastering concepts from vector algebra, students gain a powerful toolkit for solving intricate problems in physics, enabling them to accurately describe and predict the behavior of objects under various force conditions.