Circular motion of a Weightless rod

In summary, the circular motion of a weightless rod refers to the dynamics of a rod that is pivoted at one end and rotates in a circular path under the influence of external forces. The rod's lack of weight simplifies the analysis by eliminating gravitational effects on its motion. The study of this motion involves understanding angular velocity, centripetal acceleration, and the forces acting on the rod. This scenario can be modeled using principles of classical mechanics, particularly focusing on torque and rotational inertia, leading to insights into the behavior of rigid bodies in rotational motion.
  • #1
PitViper
12
1
Homework Statement
weightless rod AB of length a is free to rotate about a fixed end A. At its other end B, a particle of mass m is attached. B is connected to a ring C of mass m by an inelastic string of length a. The ring C slides smoothly along another fixed horizontal rod passing through A. Initially, points A, B, and C are aligned in a straight line with AC = 2a

The system is released from rest, and at time t, the angle CAB is θ.
Relevant Equations
(d/dt (theta)) ^ 2 = (2g)/a * (sin theta)/(1 + 4sin^2 theta)
I used law of conservation of energy to calculate (d theta/ dt)^2 (from:mgasin theta=1/2m(d theta/dt.a)^2+1/2mu^2(u is the velocity of the C ring at time=t)), but wasnt able to find u(velocity of C).Is there any relationship between the tangential velocity of B(d theta/dt.a) and velocity of C(u) that I'm missing?

0CD96F6B-B84A-4AD1-8291-1630B4B74B71.jpeg
 
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  • #2
Please post a picture showing the physical system.
 
  • #3
kuruman said:
Please post a picture showing the physical system.
038C06E6-3A1E-4E77-A7D4-947E4CE2A3A4.jpeg
 
  • #4
  • #5
PitViper said:
This is how I think It should look like.
Thank you for the drawing.

What are you asked to find? It is not clear from the statement of the problem.

Also, the drawing shows that the rod "after time = t" is horizontal. The statement of the problem says that the "weightless rod AB of length a is free to rotate about a fixed end A." Both cannot be correct. If the rod is free to rotate, it should be at some angle below the horizontal after some time has elapsed.
 
  • #6
There are two rods. C slides on a fixed, horizontal rod of unspecified length. Particle B is at the end of a massless rod that is free to rotate in a vertical plane about A. Initially, this rod is horizontal so that the ring C is a distance 2a from A.
 
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  • #7
TSny said:
There are two rods. C slides on a fixed, horizontal rod of unspecified length. Particle B is at the end of a massless rod that is free to rotate in a vertical plane about A. Initially, this rod is horizontal so that the ring C is a distance 2a from A.
Ah, I see it now. Thanks.
 
  • #8
PitViper said:
Is there any relationship between the tangential velocity of B(d theta/dt.a) and velocity of C(u) that I'm missing?
Let ##x## be the distance of C from A. Can you express ##x## in terms of ##a## and ##\theta##?
 
  • #9
TSny said:
Let ##x## be the distance of C from A. Can you express ##x## in terms of ##a## and ##\theta##?
2a cos theta?
 
  • #10
TSny said:
There are two rods. C slides on a fixed, horizontal rod of unspecified length. Particle B is at the end of a massless rod that is free to rotate in a vertical plane about A. Initially, this rod is horizontal so that the ring C is a distance 2a from A.
Yeah exactly!
 
  • #11
PitViper said:
2a cos theta?
Yes. Can you use this to get the relationship between the speeds of B and C?
 
  • #12
TSny said:
Yes. Can you use this to get the relationship between the speeds of B and C?
Hmmm I don’t quite get it
 
  • #13
How do you get velocity from position?
 
  • #14
PitViper said:
Hmmm I don’t quite get it
I can calculate the mean velocity of C by dividing displacement of C by t,but I don’t see how that can be ended up in final answer(the formula we have to prove)…
 
  • #15
TSny said:
How do you get velocity from position?
first derivative?
 
  • #16
PitViper said:
first derivative?
Oh my god dude I got it
 
  • #17
PitViper said:
Oh my god dude I got it
Thank you very much
 
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  • #18
PitViper said:
Thank you very much
EEF3AFD6-3B5F-4488-B84A-471E49A159C2.jpeg

This is it right?
 
  • #19
Can you please suggest me any problems to practice these type of questions?
 

FAQ: Circular motion of a Weightless rod

What is a weightless rod in the context of circular motion?

A weightless rod is an idealized physical object that has no mass and is often used in physics problems to simplify calculations. In the context of circular motion, it is typically considered as a rigid body that can pivot or rotate around a fixed point without experiencing gravitational forces acting on its mass.

How does a weightless rod behave when subjected to circular motion?

When a weightless rod is subjected to circular motion, it can rotate around a pivot point with its ends moving in circular paths. Since the rod is weightless, the only forces acting on it are those applied at its ends or at the pivot point, allowing for simplified analysis of the motion without the complications of gravitational forces.

What are the forces acting on a weightless rod in circular motion?

In the case of a weightless rod in circular motion, the primary forces acting on it are tension or applied forces at the ends of the rod or at the pivot point. If the rod is connected to a mass at one end, the tension in the rod will provide the necessary centripetal force to maintain circular motion.

How is angular acceleration related to the motion of a weightless rod?

Angular acceleration of a weightless rod is determined by the net torque acting on it and its moment of inertia. Since the rod is weightless, its moment of inertia is effectively zero, meaning that any applied torque will result in infinite angular acceleration. However, in practical scenarios, the rod may have some mass at the ends, which would contribute to its moment of inertia.

What are some real-world applications of the principles governing a weightless rod in circular motion?

Principles governing a weightless rod in circular motion can be applied in various fields such as engineering, robotics, and biomechanics. For example, they are relevant in the design of pendulum systems, robotic arms, and mechanisms where rotational motion is essential. Understanding these principles helps in optimizing performance and ensuring stability in such systems.

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