Understanding Driving Force in the m2 Equation: Explained by Experts

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In summary: So, in summary, the second term in the m2 equation is correct when it correctly matches the signs of the other terms. However, if it does not match the signs, then it is not a driving force.
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ANAli
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Homework Statement
(Problem 3.26; Classical Dynamics of Particles and Systems)
Figure 3-B illustrates a mass m1 driven by a sinusoidal force whose frequency is w. The mass m1 is attached to a rigid support by a spring of force constant k and slides on a second mass m2. The frictional force between m1 and m2 is represented by the damping parameter b1 and the friction force between m2 and the support is represented by b2. Construct the electrical analog of the system, and calculate the impedance.
Relevant Equations
Inserted image of equations in question. These are from the solution manual.
565-3-26p-i10.png
1654272776818.png


I am having trouble understanding why the second term in the m2 equation, b1(x'2 - x'1), is a negative term. Given that this force is the reason why m2 is moving in the first place, why is it not considered a driving force? I think that I don't have a clear understanding of what driving force means.
 
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A driving force is an external force imposed on the system that drives its motion. The frictional force is not driving because it is an internal force between different parts of your system that is not forcing the motion of the system but instead its form depends on the motion of the system.

Since it is the frictional force from 1 on 2, it is the 3rd law pair of the frictional force from 2 on 1. It must be equal in magnitude and opposite in direction.
 
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What happens if the value of b1 is so big that v1=v2?
 
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Orodruin said:
A driving force is an external force imposed on the system that drives its motion. The frictional force is not driving because it is an internal force between different parts of your system that is not forcing the motion of the system but instead its form depends on the motion of the system.

Since it is the frictional force from 1 on 2, it is the 3rd law pair of the frictional force from 2 on 1. It must be equal in magnitude and opposite in direction.
Got it. I understand. Thank you!
 
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ANAli said:
Homework Statement:: (Problem 3.26; Classical Dynamics of Particles and Systems)

I am having trouble understanding why the second term in the m2 equation, b1(x'2 - x'1), is a negative term. Given that this force is the reason why m2 is moving in the first place, why is it not considered a driving force? I think that I don't have a clear understanding of what driving force means.
One way to see if the signs make sense is to consider specific cases. For example, if ##m_1## is moving to the right (##\dot x_1>0##) and ##m_2## is at rest (##\dot x_2=0##), the top mass will push the bottom mass to the right so you would expect ##\ddot x_2>0##. In this case, the second equation reduces to ##m_2 \ddot x_2 = + b_1 \dot x_1 > 0##, which is in agreement. Similarly, if you consider the case where ##\dot x_2>0## and ##\dot x_1 = 0##, you'll see the signs of the other two terms are correct as well.
 

Related to Understanding Driving Force in the m2 Equation: Explained by Experts

1. What is the m2 equation and why is it important in understanding driving force?

The m2 equation, also known as the modified Newton's second law, is a mathematical formula that describes the relationship between mass, acceleration, and force. It is important in understanding driving force because it helps us calculate the amount of force needed to accelerate an object with a certain mass.

2. How is the m2 equation used in real-world applications?

The m2 equation is used in various real-world applications, such as designing vehicles, calculating the thrust of rockets, and understanding the motion of objects in space. It is also used in sports science to analyze the forces acting on athletes during different movements.

3. What factors affect the driving force in the m2 equation?

The driving force in the m2 equation is affected by two main factors: mass and acceleration. The higher the mass of an object, the more force is needed to accelerate it. Similarly, the greater the acceleration, the more force is required to achieve it.

4. How does understanding driving force in the m2 equation help in improving efficiency?

By understanding driving force in the m2 equation, we can optimize the amount of force needed to achieve a certain acceleration. This can lead to improved efficiency in various processes, such as designing more fuel-efficient vehicles or improving the performance of athletes.

5. Are there any limitations to the m2 equation?

While the m2 equation is a useful tool for understanding driving force, it does have some limitations. It assumes that the force applied to an object is constant and that there are no external factors affecting the motion of the object. In real-world scenarios, these assumptions may not always hold true, so the m2 equation may not provide an accurate representation of the situation.

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