Balancing a Plank: Calculating Maximum Support Distance for Equilibrium

Have you ever wondered how to determine the optimal placement of supports to ensure a plank remains balanced? This article delves into a practical problem where a plank, 5 meters in length and weighing 30 kg, is placed horizontally across two supports. We will explore how to calculate the greatest distance from each end that the supports can be placed while maintaining equilibrium.

Understanding the Problem

The problem at hand involves a plank of length L 5 meters and a mass of 30 kg. The weight of the plank can be calculated as follows:

Weight of the plank (W) Mass (m) x Acceleration due to gravity (g)

Assuming g 9.81 m/s2, the weight of the plank is:

W 30 kg x 9.81 m/s2 294.3 N (newtons)

Setting Up the Problem

Let's assume the distance from each end of the plank to the support is denoted by 'x'. Consequently, the distance between the two supports is L - 2x.

The weight of the plank acts at its center, which is at a distance of L/2 2.5 meters from either end.

Conditions for Equilibrium

For the plank to be in equilibrium, the sum of moments about any point must be zero. We can take moments about one of the supports. The moment caused by the weight of the plank about one support is:

Moment W x (2.5 - x)

The reaction force at the other support must balance this moment.

Maximum Distance Condition

The maximum distance 'x' occurs when the plank is about to tip over. This happens when the center of mass of the plank is directly above one of the supports. Mathematically, this condition can be represented as:

2.5 - x 0

Solving for x gives:

x 2.5 meters

However, since the supports are at the ends, we need to ensure that both supports can be placed symmetrically. This means the distance from each end to the support must be equal.

Conclusion

Based on the above analysis, the greatest distance from each end at which the support can be placed is:

2.5 meters

In this scenario, the supports should be placed at 2.5 meters from each end of the 5-meter plank to ensure the plank remains in equilibrium. This solution demonstrates the importance of understanding moment and torque in ensuring the stability of a system.