Pendulum with its pivot accelerating upward

[AlonsoMcLaren]

A pendulum of length, b, and bob with mass, m, is attached to a massless support moving vertically upward with constant acceleration a. Find the equation of motion.

This problem is easy with the help of Lagrangian dynamics: bθ''+(g+a)θ=0

But how to solve this problem using Newton's formulation? I really have no idea... Apparently only the tension of the string and gravity are acting on the bob. But the tension of string seems very complicated...

 

[OldEngr63]

Start by writing the position of the pendulum bob in terms of the support position, ys, the pendulum length, and the angle of the pendulum. Then differentiate to find the acceleration of the bob. From there on it is fairly straight forward F = ma, with a support motion term included.

 

[rcgldr]

But how to solve this problem using Newton's formulation?

Just combine a+g and treat that sum as the effective amount of gravity for the pendulum.

 

[OldEngr63]

Draw the FBD, show the vectors, and then it will make more sense. Just talking about it does not help very much. You need to actually work with the math to see what is going on.

 

[PJC01]

I understand the importance of using different approaches to solve a problem. In this case, while the Lagrangian dynamics approach may seem easier, it is important to also consider using Newton's formulation to fully understand the system.

To solve this problem using Newton's formulation, we can start by drawing a free body diagram of the pendulum with its pivot accelerating upward. From the diagram, we can see that the only forces acting on the bob are the tension of the string and the force of gravity. The tension of the string can be broken down into its horizontal and vertical components, which will be affected by the acceleration of the pivot.

Next, we can apply Newton's second law (F=ma) to the vertical and horizontal components of the tension force. This will give us two equations of motion, which can then be solved simultaneously to find the equation of motion for the pendulum.

It is important to note that the tension force will vary with the acceleration of the pivot, making the equations more complex. However, by using Newton's formulation, we can better understand the forces at play in this system and how they are affected by the pivot's acceleration.

In conclusion, while the Lagrangian dynamics approach may provide a simpler solution, using Newton's formulation can give us a deeper understanding of the system and its dynamics. Both approaches have their strengths and it is important to consider using both in scientific problem-solving.