Solve for Equation of Motion in Tricky Mechanics Homework

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In summary, the conversation discusses the derivation of the equation of motion for a system involving a spring and a mass connected by a bar. The challenge lies in determining the angular velocity of the bar, with suggestions and alternative approaches being sought. Someone suggests that the angular velocity can be found using the formula ω = dθ/dt.
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JohnSimpson
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Homework Statement



See attached figure. Derive the equation of motion for the following in the parameter [tex]\theta[/tex]

Homework Equations





The Attempt at a Solution



The only part that's giving me trouble is the uppermost massive bar connecting the spring to the mass. I am trying to write down its kinetic energy. I know this can be decomposed as

[tex]
T = \frac{1}{2}m\bar{v}^2 + \frac{1}{2}I_{CM}\omega^2
[/tex]

where v-bar is the center of mass velocity and omega is the angular velocity of the body about its CM. I can get the first term, but I can't begin to figure out how to get Omega in the second term. I thought about considering a small change in the position of the mass, as this of course gives rise to a rotation of this bar, but the CM moves during this and it throws me off. Suggestions on how to get Omega, or on an alternative approach would be appreciated.
 

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  • #2
Hmmm, after another attempt I still can't make heads or tails of finding Omega
 
  • #3
Hi John! :smile:
JohnSimpson said:
Hmmm, after another attempt I still can't make heads or tails of finding Omega

?? :confused: ω = dθ/dt :wink:
 

1. What is the equation of motion in mechanics?

The equation of motion in mechanics is a mathematical representation of the relationship between an object's position, velocity, and acceleration. It is usually written as s = ut + ½at², where s is the object's displacement, u is its initial velocity, a is its acceleration, and t is the time elapsed.

2. How do you solve for the equation of motion?

To solve for the equation of motion, you need to have two of the three variables - displacement, velocity, or acceleration. You can then plug these values into the equation s = ut + ½at² and solve for the missing variable. It is important to ensure that all units are consistent when solving for the equation of motion.

3. What are some common mistakes when solving for the equation of motion?

One common mistake is not converting units to a consistent system, such as using meters for displacement and kilometers per hour for velocity. Another mistake is not considering the direction of motion, which can result in negative values for displacement or velocity. It is also important to double-check calculations and use the correct formula for the given situation.

4. How do you handle tricky mechanics problems when solving for the equation of motion?

Tricky mechanics problems may involve multiple objects with different velocities or accelerations, or changing acceleration over time. In these cases, it is important to break down the problem into smaller, manageable parts and apply the equation of motion to each part separately. Drawing a diagram and labeling all given information can also help in solving tricky problems.

5. Why is it important to understand the equation of motion in mechanics?

The equation of motion is a fundamental concept in mechanics and is used to describe the motion of objects in various situations. Understanding this equation allows scientists to accurately predict an object's position, velocity, and acceleration at any given time, which is crucial in fields such as physics, engineering, and astronomy. It also helps in analyzing and solving real-world problems involving motion, such as calculating the trajectory of a projectile or the motion of planets in our solar system.

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