The Turning Effect: Torque of a Couple
What is a Couple in Physics?
In everyday language, a "couple" often means two people. In physics, it has a very specific meaning. A couple is a pair of forces that are:
- Equal in size (magnitude)
- Opposite in direction
- Acting on the same object, but not along the same line of action.
Because the two forces are equal and opposite, they cancel each other out in terms of linear motion. This means a couple will not cause an object to start moving in a straight line (translational motion). Instead, it causes only rotation (rotational motion). This pure turning effect is what we call the moment of a couple or its torque.
Imagine you are using both hands to turn the steering wheel of a car. You push up with one hand and pull down with the other. These two forces form a couple, and their combined effect makes the wheel spin.
The Formula for Torque of a Couple
The strength of the turning effect, or the torque, depends on two things:
- The size of one of the forces ($F$).
- The perpendicular distance between the lines of action of the two forces ($d$).
Formula: The torque of a couple ($\tau$) is given by:
$\tau = F \times d$
Where:
- $\tau$ (tau) is the torque of the couple, measured in newton-meters ($N m$).
- $F$ is the magnitude of one of the forces, measured in newtons ($N$).
- $d$ is the perpendicular distance between the forces, measured in meters ($m$).
Notice that the formula uses only one force, $F$, not the sum of both. Since the forces are equal, using one force and the distance between them gives us the total turning effect.
Visualizing the Perpendicular Distance
The "perpendicular distance" is the key to this calculation. It is not simply the space between where the forces are applied; it is the shortest distance between the two parallel lines on which the forces act. Think of it as the length of the "lever arm" that connects the two lines of force at a right angle.
For example, if you are using a wrench to tighten a bolt, your hands apply two forces: one up and one down. The perpendicular distance, $d$, is the diameter of the bolt head you are turning. A larger bolt head means a larger $d$, which results in a greater torque for the same applied force, making it easier to turn.
Comparing a Single Force and a Couple
It's helpful to understand the difference between the turning effect (moment) of a single force and the torque of a couple.
| Aspect | Moment of a Single Force | Torque of a Couple |
|---|---|---|
| Number of Forces | One | Two |
| Net Linear Force | Can be non-zero, may cause linear acceleration. | Always zero. Causes pure rotation only. |
| Calculation | Force $\times$ Perpendicular distance from a pivot point. | One Force $\times$ Perpendicular distance between the forces. |
| Example | Pushing a door open near the handle. | Turning a key in a lock or a water tap. |
Real-World Applications of a Couple
The torque of a couple is all around us. Here are some common examples that show its importance:
- Opening a Tap or Bottle Cap: When you turn a water tap, you apply one force forward with some fingers and a backward force with others. The perpendicular distance is the diameter of the tap. A larger tap is easier to turn because the distance $d$ is larger.
- Using a Wrench: Your hand provides a push on one side of the wrench and a pull on the other. The distance $d$ is the length across the nut or bolt head. This is why a longer wrench gives you more turning power (torque).
- Steering a Car or Bicycle: Your hands work together to create a couple on the steering wheel or handlebars, allowing you to steer smoothly.
- Pedaling a Bicycle: Your feet apply forces to the pedals. When one foot pushes down, the other foot is often pulling up slightly on the opposite pedal. This pair of forces creates a torque that turns the bicycle's crank and chainring.
A Step-by-Step Calculation Example
Let's work through a simple problem to see how the formula is applied.
Scenario: Two students are trying to turn a stubborn valve on a water pipe. They each apply a force of $30 N$ in opposite directions on opposite sides of the circular valve. The diameter of the valve is $0.2 m$. What is the torque of the couple they are applying?
- Identify the Force (F): One of the forces is $F = 30 N$.
- Identify the Perpendicular Distance (d): The distance between the lines of action of the two forces is the diameter of the valve, so $d = 0.2 m$.
- Apply the Formula:
$\tau = F \times d$
$\tau = 30 \times 0.2$
$\tau = 6 N m$
The torque of the couple is $6 N m$. This is the pure turning effect trying to rotate the valve.
Common Mistakes and Important Questions
Q: Is the torque of a couple the same as the work done?
A: No, they are different concepts. Torque measures the turning effect or the potential to cause rotation. Work done is a measure of energy transfer when a force moves an object over a distance. While both involve force and distance, torque uses the perpendicular distance between forces, while work uses the distance the force acts through. Their units are also different: newton-meters ($N m$) for torque and joules ($J$) for work.
Q: Why do we use only one force in the formula $\tau = F \times d$ and not the sum of both forces?
A: The torque of a couple is defined as the product of one force and the perpendicular distance between them. Using the sum of the forces ($F + F = 2F$) would be incorrect. The physical reason is that the turning effect is produced by the "leverage" created by the separation of the two equal and opposite forces. The formula $F \times d$ correctly captures this leverage.
Q: Can the torque of a couple be zero?
A: Yes, but only if the perpendicular distance $d$ is zero. This happens when the two equal and opposite forces act along the same straight line. In this case, they cancel each other out completely and produce no rotation at all. For a couple to exist and produce a torque, the forces must not be collinear.
Conclusion
Footnote
1. Torque ($\tau$): A measure of how much a force acting on an object causes that object to rotate. It is also called the moment of force.
2. Newton (N): The International System of Units (SI) unit of force. It is defined as the force needed to accelerate a one-kilogram mass at a rate of one meter per second squared ($1 N = 1 kg \cdot m/s^2$).
3. Line of Action: The straight line along which a force vector acts. It is an imaginary line that extends infinitely in both directions along the force vector.