Finding the force of constraint

In summary, the force of constraint exerted by the rod on the bob of a simple pendulum is not simply T = mgcos(theta). When the pendulum is swinging, the correct equation is T - Mgcos(theta) = Mv^2/L, which can also be found using the energy conservation law or by using the Lagrangian method. The correct equation for the total mechanical energy of the pendulum is mv^2/2 - mgLcos(theta) = E. The Lagrangian method avoids the forces of constraint, so it is necessary to go through force analysis to find the correct speed v.
  • #1
w3390
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Homework Statement



A simple pendulum has a mass M attached at the end of a massless rod of length L. Find the force of constraint the rod exerts on the bob.

Homework Equations





The Attempt at a Solution



It seems easy enough that the mass is constrained by the tension the rod exerts on the mass. Therefore, T = mgcos(theta). However, isn't the equation of constraint supposed to help you eliminate a variable when going through the Lagrangian to find the equation of motion?
 
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  • #2
Therefore, T = mgcos(theta).
Don't jump into conclusion too fast. This is only true when the pendulum is at rest. When it swings, it gains centripetal acceleration...
 
  • #3
Okay, so when swinging:

T - Mgcos(theta) = Mv^2/L

I still don't see how this equation helps at all.
 
  • #4
v can be found by using energy conservation law, right?
Even if Lagrangian method is applied, the result is the same.
 
  • #5
You mean like 1/2Mv^2 = Mg(L - Lcos(theta))

V = sqrt[2g(L - Lcos(theta))]
 
  • #6
Mostly like that. The exact one should be: mv^2/2 - mgLcos(theta) = E, where E is the total (initial) mechanical energy of the pendulum.
 
  • #7
So I still don't see what the point of finding this equation was. I was able to find the Lagrangian and go through and solve for the equation of motion all without having this equation.
 
  • #8
What I referred to is Newtonian method, based on force and energy analysis. Lagrangian method will still yield the same result, but you will still have to go through force analysis process since Lagrangian method avoids the forces of constraint (so here, Lagrangian method will help you find speed v, equivalent to the conservation energy equation). However, regardless of the method you use, the answer T=mgcos(theta) is wrong.
 

What is the definition of "force of constraint" in science?

The force of constraint is a term used in physics to describe the force that an object or system experiences due to its interaction with another object or system. This force is typically perpendicular to the motion of the object and is responsible for keeping the object on a specific path or trajectory.

How is the force of constraint different from other types of forces?

The force of constraint differs from other types of forces in that it is not a result of physical contact between objects. Instead, it is a result of the constraints or restrictions placed on an object's motion by its environment or by other objects. It is also often a reactive force, meaning it is only present when an object is in motion.

What are some real-life examples of the force of constraint?

One example of the force of constraint is the tension in a string or rope that is holding up a swinging pendulum. Another example is the normal force that a car experiences as it travels around a banked curve on a race track. In both cases, the force of constraint is necessary to keep the objects on their desired paths.

How can the force of constraint be calculated or measured?

The force of constraint can be calculated or measured using various mathematical equations and principles, such as Newton's laws of motion and the principles of conservation of energy and momentum. In some cases, specialized equipment such as force sensors or accelerometers may also be used to directly measure the force of constraint.

What is the significance of the force of constraint in scientific research and applications?

The force of constraint plays a crucial role in understanding and predicting the behavior of physical systems and objects. It is also essential for designing and engineering structures and machines that can withstand and utilize these forces. In fields such as robotics, biomechanics, and aerospace engineering, the force of constraint is a fundamental concept that is used to analyze and improve designs and performance.

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