Introduction to Equilibrium Systems

Understanding the principles of equilibrium systems is crucial in various fields, including physics, engineering, and architecture. This article will delve into the concept of dynamic equilibrium, specifically focusing on determining support reactions in a system involving a plank and forces at its ends.

System Description and Setup

Consider a uniform plank of mass 5 kg and length 9 meters, resting on two supports, A and B, placed 2 meters from the extreme ends. Two external forces, 305 N and 350 N, are attached to the extreme ends of the plank. The objective is to determine the support reactions at points A and B.

Equilibrium Conditions

To achieve equilibrium, two conditions must be satisfied: the sum of the forces and the sum of the torques must each be zero. This is a common approach in statics and dynamics, ensuring that the object is not only stationary but also not rotating.

Net Force Equation

The first step involves writing the net force equation, which includes the downward force of gravity/weight acting at the center of the uniform plank.

Sum of Forces:

[ 311.5 , text{N} 59.87 , text{N} 5 times 9.81 times frac{9}{2} - 305 , text{N} - 350 , text{N} ]

[ 311.5 , text{N} 233.5 , text{N} 231.375 , text{N} - 305 , text{N} - 350 , text{N} ]

[ 311.5 , text{N} 311.5 , text{N} ]

Net Torque Equation

Next, we need to write the net torque equation by selecting a pivot point. We will use support A as the pivot point to eliminate one of the unknowns.

Sum of Torques about Point A:

[ tau_A 0 ]

[ tau_A 305 , text{N} times 7 , text{m} tau_{B} times 5 , text{m} - 5 times 9.81 times frac{9}{2} times frac{9}{9} , text{m} ]

[ 0 305 , text{N} times 7 , text{m} - 5 times 9.81 times frac{9}{2} times frac{9}{9} , text{m} 392.5 , text{N} times 5 , text{m} ]

[ 0 2135 , text{N} cdot text{m} - 220.65 , text{N} cdot text{m} 1962.5 , text{N} cdot text{m} ]

[ 0 3877 , text{N} cdot text{m} ]

Calculating Support Reactions

By solving the net force and torque equations, we can determine the support reactions at points A and B.

Support Reactions at A and B:

[ T_A 311.5 , text{N} ]

[ T_B 392.5 , text{N} ]

Conclusion

In this dynamic equilibrium system, the support reactions at points A and B have been successfully determined using the principles of force and torque balance. Understanding these principles is essential for solving similar problems in structural and mechanical engineering.

References

1. Halliday, D., Resnick, R., Walker, J. (2014). fundamentals of physics. John Wiley Sons.

2. Marion, J. B., Thornton, S. T. (2014). classical dynamics of particles and systems. Elsevier.