How Can Conservation Laws Simplify Pendulum Motion on a Rotating Shaft?

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In summary, the problem is about using conservation laws to analyze the motion of a pendulum mounted on a vertical, rotating shaft. The shaft is free to rotate in its bearings and one degree of freedom must be eliminated. The proposed solution involves using the Energy principle and deriving the energy expression, but the specific variable to derive on is still unknown.
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abotiz
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Homework Statement



A pendulum is mounted on a vertical, rotating
shaft by means of a hinge. The shaft is free
to rotate in its bearings. Use conservation laws to
eliminate one degree of freedom and simulate the
motion of the other one. Analyse and comment
using mechanical terms.


https://www.physicsforums.com/attachment.php?attachmentid=18687&d=1241008388




The Attempt at a Solution



I started out with the Energy principle: E = constant = Kinetic + Potential
U = Potential------I = Moment of inertia
K= Kinectic------w = angular velocity
m = mass------v = velocity
L = Length------ θ = Angle

K = ½(*I*w2+*m*v2)

U = -½*m*g*L*cos θ

And i have no idea what to do next, i think i perhaps should derive the energy expression, but i don't know what to derive on, time,angle?

Thank you!
 
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  • #2
Sorry, problem with image

Here it is


Pend.jpg
 

FAQ: How Can Conservation Laws Simplify Pendulum Motion on a Rotating Shaft?

How do I calculate the period of a swinging pendulum?

The period of a swinging pendulum can be calculated using the equation T = 2π√(L/g), where T is the period in seconds, L is the length of the pendulum in meters, and g is the acceleration due to gravity (9.8 m/s2). This equation assumes that the amplitude of the pendulum's swing is small (less than 15 degrees).

What factors affect the period of a swinging pendulum?

The period of a swinging pendulum is affected by its length, mass, and the force of gravity. The longer the pendulum, the longer the period. Similarly, a heavier pendulum will have a longer period. The force of gravity also plays a role, with a higher force resulting in a shorter period.

How can I determine the length of a pendulum to achieve a desired period?

To determine the length of a pendulum needed to achieve a desired period, you can rearrange the equation T = 2π√(L/g) to solve for L. The equation would become L = (T/2π)2 * g. Plug in the desired period and the value of gravity to calculate the necessary length.

What is the difference between a simple pendulum and a compound pendulum?

A simple pendulum is a mass suspended from a fixed point by a string or rod. It exhibits simple harmonic motion, meaning its period remains constant regardless of its amplitude. A compound pendulum, on the other hand, has a distributed mass and a complex shape. Its period can vary depending on its amplitude.

Can the period of a pendulum be affected by air resistance?

Yes, air resistance can affect the period of a pendulum. As the pendulum swings, it moves through the air, causing air resistance. This resistance acts as a damping force, reducing the amplitude of the pendulum's swing. As a result, the period of the pendulum will be slightly longer than if there were no air resistance.

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