Active force of spring on mass (D'alemberts)

In summary, the question being discussed is the effect of eliminating the mass m1 (m1=0) on the active forces (Fa) in a system with two masses and a spring. The active force on m1 is from the spring and does not depend on the mass, but the equations of motion do. When m1 is eliminated, the set of D'alemberts reduces and the remaining equation of motion is simplified. However, it is unclear how the active force from the spring affects the remaining equation of motion. Further clarification and understanding is needed.
  • #1
buildingblocs
17
1
I am trying to understand the effect that eliminating the mass m1 (m1 = 0) has on the active forces (Fa). I have gone through a scenario where m1 is taken into consideration (refer to uploaded images). The active force on m1 is from the spring and does not have m1 in the expression (Fs1 = (l-r)k), so I did not think that there would be any change. However the set of D'alemberts reduces if m1 is eliminated, so how does active force from the spring effect the remaining equation of motion.

If more info or clearing up is required please do not hesitate to ask. Any help would be much appreciated.

Cheers.
 

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  • #2
Can you write it up there? The attachments are hard to read and their wrong orientation does not help either.

I don't understand what you want to test/show.
The force of the spring at a specific location does not depend on the mass, but the equations of motion certainly do because acceleration is force divided by mass. This also means you cannot expect a physical answer if you set the mass to zero. If you have a spring without mass attached, you still have the mass of the spring (which is often neglected).
 
  • #3
Figure1.png

figure1.

In figure1, the generalised variables are q and θ, and there are two masses (mass of the spring is neglected). The active forces are Fa1 = Fspring1 (horizontal direction) and Fa2 = Fweight (vertical direction). Therefore it follows that there should be four parts for the equations of motion:
(Fa1 - m1a)∂r1/∂q + (Fa1 - m1a)∂r1/∂θ + (Fa2 - m2a)∂r2/∂q + (Fa2 - m2a)∂r2/∂θ = 0 [eq1]

this is simplified to:
(Fa1 - m1a)∂r1/∂q + (Fa2 - m2a)∂r2/∂q + (Fa2 - m2a)∂r2/∂θ = 0 [eq2]

as ∂r1/∂θ=0

My query is what happens when m1 is eliminated from the system (m1 = 0)
The is now two generalised variable and only one mass:
(Fa2 - m2a)∂r2/∂q + (Fa2 - m2a)∂r2/∂θ = 0
[eq3]


figure2.png

figure 2.

Now consider figure2 (Spring is removed and m1=0)
The equation of motion of system in figure2 is the same as that for eq3, where the spring was included. This intuitively does not appear correct, and I wish to understand how to take into account the effect of spring on the remaining system when m1 =0;
 
  • #4
If you set m1 to zero, you cannot ignore the forces on r1 - they are still part of the mechanics. The only difference is the exact cancellation of the force from the spring and the horizontal component of the force in the beam (the experiment does not work properly with a string of course)
 

1. What is the active force of a spring on a mass?

The active force of a spring on a mass, also known as D'alembert's force, is the force exerted by a spring on a mass when it is in motion. It is equal to the negative of the mass times its acceleration.

2. How is D'alembert's force calculated?

D'alembert's force is calculated by multiplying the mass of the object by its acceleration and then taking the negative of that value. This gives the force that the spring exerts on the mass in the opposite direction of its motion.

3. What is the relationship between the active force and the displacement of the spring?

The active force of a spring on a mass is directly proportional to the displacement of the spring. This means that as the spring is stretched or compressed, the force it exerts on the mass will also change in the same direction.

4. How does the mass of an object affect the active force of a spring?

The mass of an object has a direct impact on the active force of a spring. The greater the mass, the greater the force that the spring will need to exert to move the object at a certain acceleration.

5. What are some real-life examples of D'alembert's force?

D'alembert's force can be observed in various real-life scenarios, such as a spring-loaded door closing, a person jumping on a trampoline, or a car hitting a speed bump. In all of these situations, the spring is exerting a force on the object in motion to counteract its acceleration.

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