Lagrangian mechanics, simple pendulum

In summary: If you just begin the problem that the disk rotates on the other side, you get the same answer as textbook.
  • #1
YauYauYau
8
0

Homework Statement


A simple pendulum of length ξ and mass m is suspended from a point on the circumference of a thin massless disc of radius α that rotates with a constant angular velocity ω about its central axis as shown in Figure. Find the equation of motion of the mass m.
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Homework Equations


L = T - V

The Attempt at a Solution


Firstly, I know I should find its x' and y' which representing its velocity.
i.e. T - V = 1/2 m ( x_dot 2 + y_dot 2 ) - mgy

However, I don't know where I should start with to find their x and y.

Thanks.
 
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  • #2
YauYauYau said:

The Attempt at a Solution


Firstly, I know I should find its x' and y' which representing its velocity.
i.e. T - V = 1/2 m ( x_dot 2 + y_dot 2 ) - mgy

However, I don't know where I should start with to find their x and y.

Thanks.
first task would be to define the generalized coordinates q and velocities qdots and then proceed for defining the T and V.
your disk is in which plane and which is axis of rotation of the disk.
 
  • #3
drvrm said:
first task would be to define the generalized coordinates q and qdots and then proceed for defining the T and V.

But how can I define the generalized coordinates q and qdots?
 
  • #4
YauYauYau said:
But how can I define the generalized coordinates q and qdots?

define usual coordinates as degrees of freedom permits -write any constraining equations...which are relations between coordinates or velocities.
then the gen. coordinates can be defined.
see your textbook
 
  • #5
YauYauYau said:
But how can I define the generalized coordinates q and qdots?

your pendulum is hanging from a disk- so it will be at length L from the disk suppose you place your origin of coordinates at the centre of disk
and the axes X,Y,Z so the bob will lie at -z,x,y but as disl starts rotating the bob will start rotating in a circle and that circle will be a raised one , so at any instant the bob will be at x',y'z' .
the equation of constraint can be that sum of the squares of three coordinates will be equal to length square + rad of the disk squared.
if you choose an angle made by the thread with vertical the cosine of the angle will be z' /L... similarly other relations can follow and your degrees of freedom will be reduced- the motion may be described by one angle and its time rate of change.
 
  • #6
drvrm said:
your pendulum is hanging from a disk- so it will be at length L from the disk suppose you place your origin of coordinates at the centre of disk
and the axes X,Y,Z so the bob will lie at -z,x,y but as disl starts rotating the bob will start rotating in a circle and that circle will be a raised one , so at any instant the bob will be at x',y'z' .
the equation of constraint can be that sum of the squares of three coordinates will be equal to length square + rad of the disk squared.
if you choose an angle made by the thread with vertical the cosine of the angle will be z' /L... similarly other relations can follow and your degrees of freedom will be reduced- the motion may be described by one angle and its time rate of change.

I try to begin the question with place origin of coordinates at the centre of the disk (x,y)
With the parameters of figures given,
for x, because ξsinθ ( the height of triangle formed by bob ) is longer than that of radius,
x = ξsinθ - a cos ωt
similarly, y = ξcosθ - a sin ωt

However, the answer is x = a cos ωt + ξ sin θ, y = a sin ωt - ξcosθ
Am I missing something?

If I just begin the problem that the disk rotates on the other side, I get the same answer as textbook.
Last night, I spent a night to watch youtube mechanics and vector teaching videos and read over my textbook but still I could not get the answer.
I am new to Mechanics. sorry for any annoying questions :'(
 

1. What is Lagrangian mechanics?

Lagrangian mechanics is a mathematical framework used for analyzing the dynamics of a system. It is based on the principle of least action, which states that the path taken by a system between two points is the one for which the action (a measure of the system's energy) is minimized.

2. How is Lagrangian mechanics different from Newtonian mechanics?

Lagrangian mechanics differs from Newtonian mechanics in that it takes into account the total energy of a system, including both kinetic and potential energy. It also uses generalized coordinates instead of specific forces, making it more suitable for analyzing complex systems.

3. What is a simple pendulum?

A simple pendulum is a weight suspended from a pivot point by a thin, light string or rod. It is used as a model system to study the principles of oscillatory motion and is often used in introductory physics courses.

4. How is a simple pendulum analyzed using Lagrangian mechanics?

In Lagrangian mechanics, the motion of a simple pendulum is described using the pendulum's position and velocity, rather than the forces acting on it. The Lagrangian equation is used to determine the pendulum's equations of motion and predict its behavior.

5. What factors affect the motion of a simple pendulum?

The motion of a simple pendulum is affected by the length of the string, the mass of the weight, and the acceleration due to gravity. It also depends on the initial conditions, such as the amplitude and velocity of the pendulum's swing.

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