Suppose equal weights are put on the two pans of a beam balance, the arm is kept at an angle with the horizontal and released. Is the torque of the two weights about the middle point (point of support) zero? If so, why does the arm rotate and finally become horizontal?

When equal weights are placed on both pans of a beam balance, and the arm is tilted at an angle before being released, it might seem at first that the torques about the pivot point (the middle point) would cancel each other out, leading to no movement. However, this isn't the case when the arm is angled. Let's break this down to understand why the arm rotates and eventually comes to a horizontal position.

The Concept of Torque

Torque is a measure of the rotational force acting on an object about a pivot point. It depends on two main factors: the force applied and the distance from the pivot point (also known as the lever arm). The formula for torque (τ) can be expressed as:

• τ = r × F × sin(θ)

Here, r is the distance from the pivot to the point where the force is applied, F is the force (in this case, the weight), and θ is the angle between the lever arm and the line of action of the force.

Analyzing the Situation

In your scenario, when the beam is tilted, both weights exert a downward force due to gravity. Since they are equal, one might think they balance each other out. However, the key lies in the angle of the arm:

• When the arm is tilted, the distance from the pivot point to each weight (the lever arm) is not the same in terms of effective torque.
• The angle of the arm affects the sine component in the torque equation, which means that even though the weights are equal, the torque they produce about the pivot is not zero.

Why Does the Arm Rotate?

When the beam is released, the weight on the lower side of the angle has a greater effective torque compared to the weight on the higher side. This is because:

• The distance from the pivot to the lower weight is greater than the distance to the higher weight when the arm is tilted.
• The angle at which the force acts also plays a role; the effective lever arm changes as the angle changes.

As a result, the torque produced by the lower weight is greater than that of the upper weight, causing the arm to rotate downward toward the horizontal position.

Final Position

The arm continues to rotate until it reaches a horizontal position because, at this point, the torques from both weights become equal again. In the horizontal position, the lever arms are equal, and the gravitational forces act directly downward, leading to a state of equilibrium.

Conclusion

In summary, while the weights are equal, the angle of the beam creates a situation where the torques are not balanced due to differing lever arms and angles. This imbalance causes the arm to rotate until it reaches a horizontal position, where equilibrium is restored. Understanding these principles of torque and balance is crucial in physics, as they apply to many real-world scenarios beyond just beam balances.