Pendulum with its pivot accelerating upward

In summary, the conversation discusses how to find the equation of motion for a pendulum with a massless support moving vertically upward with constant acceleration. The problem can be solved using Lagrangian dynamics or Newton's formulation, but the latter requires breaking down the forces acting on the bob, including gravity and the tension of the string. It is suggested to start by writing the position of the bob in terms of the support position and angle, then differentiate to find the acceleration. Drawing a free body diagram and working with the math will provide a better understanding of the problem.
  • #1
AlonsoMcLaren
90
2
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...
 
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  • #2
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.
 
  • #3
AlonsoMcLaren said:
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.
 
  • #4
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.
 
  • #5


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.
 

FAQ: Pendulum with its pivot accelerating upward

What is a pendulum with its pivot accelerating upward?

A pendulum with its pivot accelerating upward is a physical system consisting of a weight (known as the pendulum bob) attached to a rod or string that is suspended from a fixed point. The pivot point, or point of suspension, is being accelerated upward, causing the pendulum to move in a circular motion.

How does the acceleration of the pivot affect the motion of the pendulum?

The acceleration of the pivot affects the motion of the pendulum by changing the direction and speed of the pendulum's movement. As the pivot point moves upward, the pendulum will swing in an elliptical path rather than a straight line. The acceleration of the pivot also affects the period (the time it takes for the pendulum to complete one full swing) and the amplitude (the maximum angle of the pendulum) of the pendulum's motion.

What factors influence the motion of a pendulum with its pivot accelerating upward?

The motion of a pendulum with its pivot accelerating upward is influenced by several factors, including the acceleration of the pivot, the length of the pendulum, the mass of the pendulum bob, and the initial angle of the pendulum. These factors can affect the period and amplitude of the pendulum's motion, as well as the shape of the path it follows.

What is the relationship between the acceleration of the pivot and the period of the pendulum?

The acceleration of the pivot and the period of the pendulum have an inverse relationship. This means that as the acceleration of the pivot increases, the period of the pendulum decreases, and vice versa. This is because a higher acceleration of the pivot causes the pendulum to move faster, reducing the time it takes to complete one full swing.

How does the acceleration of the pivot affect the energy of the pendulum?

The acceleration of the pivot affects the energy of the pendulum by changing the potential and kinetic energy of the system. As the pivot moves upward, the potential energy of the pendulum increases, while the kinetic energy decreases. This results in a transfer of energy between potential and kinetic forms, causing the pendulum to oscillate at a different rate than it would with a stationary pivot.

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