Classical mech non-inertial frame bead on a rotating ring

In summary, The problem involves finding the equation of motion of a bead threaded on a circular hoop while in a rotating frame. The equation of motion is given by d2θ/dt2 = (ω2cosθ - g/R)sinθ, and it is shown to agree with eq 7.69. Using a free body diagram, the equilibrium position is explained by the forces of gravity, centrifugal force, and coriolis force. The centrifugal force is given by mRω2sinθ, while the coriolis force is -2mvΩcosθ. The effective force is equal to mR(d2θ/dt2) and can be solved using Lagrange method. However, understanding the concept
  • #1
Liquidxlax
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Homework Statement



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Consider the bead threaded on a cicular hoop of example 7.6 (pg 260), working in a frame that rotates with the hoop. find the equation of motion of the bead, and check that your result agrees with eq 7. 69. Using a free body diagram explain the result 7.71 for equilibrium positions

Homework Equations



d2θ/dt2 = (ω2cosθ - g/R)sinθ 7.69

θo = ±arccos(g/ω2R) 7.72

The Attempt at a Solution



In the inertial frame there is going to be a centrifugal force coriolus force and force of gravity

Feff = Fg + Fcf + Fcor

by the diagram

Fcf = mRω2sinθ Not sure about the direction

( I'm thinking it wouldn't really be in the r hat direction. more like Rcosθ. sounds redundant but I'm not sure how to explain it.)

Fcor = -2mvΩcosθ because it is in the southern hemisphere so it would deflect left which would oppose the gravitational force

I assume Feff = mR2[d2θ/dt2]



I can solve this problem by lagrange method, but I'm not fully understanding this non-inertial ref frame.
 
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  • #2
Suprisingly enough i actually did it right

Feff is mR(d2θ/dt2) = mRΩ2sinθ(rho_hat) -mgRsinθ

where rho_hat is cosθ

Deduced this by saying r' = r therefore R has to equal zero, but capital R is not the same as the radius in the equation above
 

1. What is "Classical mech non-inertial frame bead on a rotating ring"?

Classical mechanics refers to the branch of physics that studies the motion of objects under the influence of forces. In this scenario, a non-inertial frame refers to a reference frame that is accelerating or rotating. The bead on a rotating ring refers to an object moving along the surface of a ring that is rotating around a central axis.

2. What is the significance of studying this scenario?

Studying this scenario helps us understand the effects of non-inertial frames on the motion of objects. It also allows us to apply the principles of classical mechanics to real-world situations, such as the motion of objects on spinning rides or in rotating machinery.

3. How does the motion of the bead change in a rotating reference frame?

In a rotating reference frame, the bead experiences a centrifugal force that pulls it away from the center of rotation. This force acts in the opposite direction of the centripetal force that keeps the bead moving in a circular path.

4. What is the difference between an inertial and non-inertial reference frame?

An inertial frame is a reference frame in which Newton's laws of motion hold true and the object's velocity and acceleration remain constant. A non-inertial frame is one in which the velocity and acceleration of an object change due to forces acting on it, such as in the case of rotating or accelerating frames.

5. How do we calculate the motion of the bead in this scenario?

To calculate the motion of the bead, we can use the equations of motion in a non-inertial reference frame, which take into account the centrifugal force and the Coriolis force (due to the rotation of the reference frame). These equations can be derived from Newton's laws of motion and the principles of classical mechanics.

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