How to find max deflection of spring between 2 masses ?

In summary, the conversation discusses a problem involving two masses connected by a spring and a force applied to one of the masses. The maximum displacement of the spring is determined using the acceleration of the center of mass and the individual accelerations of each mass due to the spring force. The solution involves finding the endpoints of the system's oscillation in a parabola and considering the external force's effect on the spring potential. The question of whether a1 is equal to a2 is raised, but not directly addressed.
  • #1
bksree
77
2
Hi
I came across a problem as below :
2 masses M1 and M2 are connected by a spring and kept on a frictionless table horizontally. A force F is applied to M2. What is the maximum displacement of the spring ?

The acc of the COM of the system is a = F/(m1 + m2).
However, each mass will move with different acceleration because of the spring between them. If x1 and x2 are the displacements at M1 and M2, the force exerted by the spring on each mass will be k(x1-x2).
FBD of M1 gives
k(x1-x2) = M1 a1
FBD of M2 gives
F-k(x1-x2) = M2 a2

I'm not clear how to proceed from here. The ans is
2M1 * F / (M1 + M2)

Cany anypone explain ?

TIA
 
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  • #2
I would split the system in the center of mass movement and the movement of the individual objects. The whole system accelerates with a=F/(m1+m2). In this system, an effective force of F-a*m1 accelerates m1 inwards and a*m2 accelerates m2 inwards. As a(m1+m2)=F, both have the same magnitude - this is just a cross-check.

Let x be the distance relative to the springs at rest (x=0), with positive x as larger distance. The spring potential is now V(x)=1/2*k*x^2 and the external force adds a potential of -2*a*m2*x. The system will oscillate in this parabola, with 0 potential as endpoints. One is x=0, can you find the other one?
 
  • #3
Thanks for the reply. But why should a1 be equal to a2 ? The acceleration of COM is a = F/(m1 + m2)
Hence a = (m1a1 + m2a2) / (m1+m2).
Since x1, m1 and F1 are not equal to x2, m2 and F2, how can we conclude that a1 = a2 ?

TIA
 
  • #5


To find the maximum deflection of the spring between the two masses, we can use the concept of equilibrium. At maximum deflection, the spring will be stretched to its maximum length and the system will come to a stop. This means that the net force on each mass will be zero.

To find this point of equilibrium, we can set up equations for the forces acting on each mass. For mass M1, the force from the spring will be k(x1-x2) and the force from the applied force F will be M1a1. Similarly, for mass M2, the force from the spring will be k(x1-x2) and the force from the applied force F will be M2a2.

At equilibrium, the net force on each mass will be zero, so we can set these equations equal to zero and solve for the displacement x1-x2. This will give us the maximum deflection of the spring.

To simplify the equations, we can use the acceleration of the center of mass of the system, a = F/(m1+m2), and substitute it into the equations for a1 and a2. This will give us:

k(x1-x2) = M1(F/(m1+m2))
k(x1-x2) = M2(F/(m1+m2))

Solving for x1-x2, we get:

x1-x2 = (2M1*M2*F)/(k*(m1+m2)^2)

This is the maximum deflection of the spring between the two masses, which matches the answer provided. I hope this explanation helps clarify the process for finding the maximum deflection of the spring in this system.
 

FAQ: How to find max deflection of spring between 2 masses ?

1. How do you calculate the maximum deflection of a spring between two masses?

To calculate the maximum deflection of a spring, you need to know the spring constant, the mass of the object attached to the spring, and the distance between the two masses. The formula for maximum deflection is: max deflection = (mass x gravity x distance^2) / (2 x spring constant).

2. What is the significance of the spring constant in calculating maximum deflection?

The spring constant is a measure of how stiff or flexible a spring is. It is the force required to compress or stretch a spring by a certain distance. In the formula for maximum deflection, a higher spring constant means the spring will have less deflection, while a lower spring constant will result in more deflection.

3. Can the mass of the objects attached to the spring affect the maximum deflection?

Yes, the mass of the objects attached to the spring will affect the maximum deflection. The heavier the objects, the more force will be exerted on the spring, resulting in a greater deflection. However, the distance between the two masses also plays a role in determining the maximum deflection.

4. What happens to the maximum deflection if the distance between the masses is doubled?

If the distance between the masses is doubled, the maximum deflection of the spring will also double. This is because the distance is squared in the formula for maximum deflection, meaning it has a direct impact on the amount of deflection.

5. How can I experimentally determine the maximum deflection of a spring between two masses?

To experimentally determine the maximum deflection of a spring, you will need to set up an apparatus with the spring attached to two masses. Measure the distance between the masses and record the mass of the objects. Then, slowly add weight to one of the masses and measure the deflection of the spring. Repeat this process with different masses and distances to get a range of data points, and then use the formula to calculate the maximum deflection.

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