Point Mass Inside Hoop: Parametric Equation & Euler-Lagrange E.o.m.

  • Thread starter Oskar Paulsson
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In summary, the hoop has three degrees of freedom (roll, translation, and rotation) and the point mass has only translational and rotational energy.
  • #1
Oskar Paulsson
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Homework Statement


A point mass is constrained to move along inside a hoop.
The hoop can roll frictionless and without slipping on a horizontal plane along a fixed direction.
At some initial time t = 0 the hoop is at rest and the particle is at the top of the hoop with velocity vtop

Homework Equations


Suitable parametric equation for the point mass and compute the Euler-Lagrange E.o.m.

The Attempt at a Solution


First some drawings.
osemmo.png


Since the "ball" is a point mass I consider its own radius to be negliable, hence it only has translational energy.
The velocity of the ball itself:
$$ v = r \alpha $$
$$ \alpha = \frac{d\theta}{dt} $$
The hoop has clockwise rotational motion
$$\omega = \frac{d\phi}{dt}$$

The point mass must have some parametric co-ordinate
$$ x(r,\theta) $$
which I'm having trouble finding.
My reasoning on this is as follows; the point can be described in terms of height about the horizontal line of the hoop
$$ rsin(\theta) $$
With an additional term for the horizontal displacement of the hoop:
$$ r\omega t$$
$$ x(r,\theta) = rsin(\theta) + r\omega t $$
then
$$ \alpha = \dot{x}(r,\theta)$$

Is this right?
 
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  • #2
Oskar Paulsson said:
The velocity of the ball itself:
This is not the total velocity of the ball.

ω is not constant, you cannot calculate the horizontal displacement like that.
 
  • #3
Welcome to PF, Oskar.

As mfb pointed out, the hoop will have a complicated motion as it rolls in response to the time varying forces exerted on it by the moving particle and the friction from the surface.

You might want to step back and first consider how many degrees of freedom the entire system has. Then judiciously choose a set of independent, generalized coordinates for specifying the configuration of the system. The number of generalized coordinates should equal the number of degrees of freedom. Then see if you can express the kinetic and potential energies of the system in terms of the generalized coordinates.
 

1. What is a point mass inside hoop and why is it important in physics?

A point mass inside hoop refers to a simplified model of a particle moving along a circular path inside a hoop structure. This model is important in physics because it allows for the analysis of the motion of a particle in a constrained environment, which has applications in various fields such as mechanics, fluid dynamics, and quantum mechanics.

2. What is the parametric equation for a point mass inside hoop?

The parametric equation for a point mass inside hoop is given by x = Rcos(ωt), y = Rsin(ωt), and z = h, where R is the radius of the hoop, ω is the angular velocity, t is time, and h is the height of the hoop.

3. How is the Euler-Lagrange equation of motion used in the analysis of a point mass inside hoop?

The Euler-Lagrange equation of motion is used to determine the path that a point mass will take inside a hoop. This equation takes into account the kinetic and potential energy of the particle, as well as any external forces acting on it, to find the path of least action that the particle will follow.

4. Can the parametric equation and Euler-Lagrange equation be used for any type of hoop?

Yes, the parametric equation and Euler-Lagrange equation can be used for any type of hoop as long as the appropriate values for R, ω, and h are substituted into the equations.

5. Are there any real-world applications of the analysis of a point mass inside hoop?

Yes, there are several real-world applications of the analysis of a point mass inside hoop. This includes the study of planetary motion, the movement of particles in a centrifuge, and the behavior of electrons in a magnetic field. It also has applications in sports, such as analyzing the motion of a ball in a basketball hoop or the trajectory of a tennis ball inside a racket.

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