Calculating generalized force in two different ways

In summary, the problem involves a ladder of length L and mass m leaning against a frictionless wall and floor. The ladder is initially at an angle θ and is sliding down the floor until the left end loses contact with the wall. The goal is to express the kinetic energy in terms of θ and dθ/dt, and to find the virtual work and generalized force. The equations and attempts at solutions are provided in the attached images. However, there are discrepancies in the calculated generalized force and further clarification or hints are needed to identify where the mistake was made.
  • #1
amjad-sh
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Homework Statement


a ladder of length L and mass m is leaning against a wall as shown in the figure below. assuming the wall and the floor are frictionless, the ladder will slide down the floor and until the left end loses contact with the wall. Before the ladder loses contact with the wall there is one degree of freedom θ.
1.Express the kinetic energy interms of θ and dθ/dt.

2.find the virtual work and the generalized force.

Homework Equations


Note: the bold symbols are related to vectors.
T=1/2 ×i=1∑i=M (mi×dsi/dt⋅dsi/dt)=T(q1,...,qn,dq1/dt,...,dqn/dt).⇒⇒⇒formula of kinetic energy. (1)
M is all parts of the system.
δw=k∑×(i∑Fi*∂ri/∂qk)δqk.⇒⇒⇒formula of virtual work. (2)
k is the number of degrees of freedom,and qk is a degree of freedom
i∑Fi*∂ri/∂qk=Fk(generalized force). (3)

Fk=d/dt(∂T/∂(dqk/dt))-∂T/∂qk.(4)

The Attempt at a Solution


[/B]I used Cartesian coordinates, the part related to the ground is the x-axis and how the ladder is high from the ground is the y axis. Let the origin be the point of contact of the ladder with the ground and "s" is how each infinitesimal part is far from the origin.

1.Let "h" be the distance between the point of contact of the ladder with the ground and the wall, let "y" the distance between the point of contact of the ladder with the wall and the ground.
The vector Si=hii+yij.
dsi/dt=dhi/dti+dyi/dtj.
h=cosθs.
y=sinθs.
si.si=(dθ/dt)2 *(sinθ)2*s2 +(dθ/dt)2*(cosθ)2*s2
(dsi/dt)2=s2 *(dθ/dt)2
T=0∫L (dT)=0∫L (1/2*dm*(ds/dt)2)=0.5*0∫L(M/L*ds*s2*(dθ/dt)2)=0.5*(dθ/dt)2*M/L*0∫Ls2*ds=1/6*(dθ/dt)2*ML2

2. I calculated here the generalized force.

∂ri/∂θ=∂h/∂θi+∂y/∂θj=-(dθ/dt)sinθsi+(dθ/dt)cosθsj

Using contious form of equation (3) we get :
Fk=(intergrate from zero to L) ∫dmgj.(-(dθ/dt)*sinθ*si +dθ/dt*cosθ*sj)
I get finally Fk=Mg(dθ/dt)*cosθ*(L/2).

when I use equation (4) to get the generalized force I get different answer, so is there something wrong in my solution?
Capture.GIF
 
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  • #2
I'm afraid your post is too hard to read the way it is set out, so there is unlikely to be much help offered. I suggest you read the physicsforums LaTeX primer about how to set out formulas using LaTeX, then you'll be able to rewrite the post in a way that's easy to read, and you're likely to get much more help.

Good luck!
 
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Likes amjad-sh
  • #3

I rewrote my question, but I used my handwriting to write my attempt to the solution.I will use Latex in my later posts.

Homework Statement


a ladder of length L and mass m is leaning against a wall as shown in the figure below. assuming the wall and the floor are frictionless, the ladder will slide down the floor and until the left end loses contact with the wall. Before the ladder loses contact with the wall there is one degree of freedom θ.
1.Express the kinetic energy interms of θ and dθ/dt.

2.find the virtual work and the generalized force.

Homework Equations


They are written in the papers uploaded below.

3. The Attempt at a Solution


I calculated the generalized forced in different ways, but unfortunately I got two different answers.
So if anybody can show to me a hint or tell me where I missed.
My attempts are written in the uploaded pictures below.
dm=M/L*ds.
I integrated from zero to L in all the integrals written.
and it is dm in the integral of page 4 not m.
 

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1. What is generalized force and why is it important in scientific calculations?

Generalized force is a concept used in physics to describe the effect of a force on a system. It takes into account both the magnitude and direction of the force, as well as the position and orientation of the system. This is important in calculations because it allows for a more accurate representation of the dynamics of a system.

2. How do you calculate generalized force in two different ways?

There are two main ways to calculate generalized force: using Newton's second law of motion and using Lagrange's equations. Newton's second law involves multiplying the mass of an object by its acceleration, while Lagrange's equations use the concept of virtual work to determine the generalized force. Both methods can yield the same result, but they are based on different principles.

3. What is the difference between conservative and non-conservative generalized forces?

Conservative generalized forces are those that can be derived from a potential energy function, while non-conservative forces cannot. This means that conservative forces do not dissipate energy, while non-conservative forces do. In terms of calculations, conservative forces can be easier to work with because they have simpler mathematical expressions.

4. Can generalized force be negative?

Yes, generalized force can be negative. This means that the force is acting in the opposite direction of the displacement of the system. For example, if a force is applied to an object in the negative x-direction, but the object is moving in the positive x-direction, the generalized force would be negative.

5. How does calculating generalized force help in understanding the behavior of complex systems?

Calculating generalized force is an important step in understanding the dynamics of a system and predicting its behavior. By taking into account all the forces acting on a system, including both internal and external forces, we can better understand how the system will respond to different conditions and interactions. This is especially useful in complex systems where individual components may have multiple forces acting on them.

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