Spring attached to string on one side, mass on the other.

[mrdoe]

This is a problem I came up with myself, so it's not homework and so I didn't post in that forum.

Suppose a mass m1 rests on a frictionless table. m1 is directly connected to a massless ideal spring at equilibrium length of spring constant k which is connected to a string going over a pulley of mass m3 and radius r (and rot. inertia 0.5m3r^2), which is connected after the pulley to a mass m2.

Determine the equation for the motion of mass m1 when m2 is released. (dig. attached).

How would you go about doing this problem? I can't seem to get started because I can't figure out how to find the length the spring will initially be extended.

relevant equations:
F = -kx
Conservation of energy, probably.
All the angular/linear kinematics equations.

edit:sorry reread the rules just then, mods please move this to the homework forum (i thought the restriction didn't apply to independent study)

 

[tiny-tim]

Hi mrdoe!

This is a problem I came up with myself …
…
Determine the equation for the motion of mass m1 when m2 is released. (dig. attached).

How would you go about doing this problem? I can't seem to get started because I can't figure out how to find the length the spring will initially be extended.

Use conservation of energy.

And since you came up with this problem yourself, I'm surprised you've forgotten that you decided that "The spring is held at equilibrium before mass 2 is released" (in the diagram) … which I assume means that the spring is held at its natural length.

 

[jack action]

Since it is a frictionless table, the spring will never deform.

Imagine the problem another way: hold m1 over m2 in the air (no pulley), linked by the same string and spring. release them both under acceleration g. What happen? Both fall at the same velocity and the spring never get stretched, because there are no opposite forces.

You need friction to stretch the spring. And, in that case, the equation of motion would be easy since m2 would move while m1 is at rest. m1 will start to move when the spring force will overcome the static friction. At this point, motion of m1 will be the same as motion m2 with the extra distance created by the stretched spring.

 

[mrdoe]

Thanks, it makes sense now.
Intuitively the initial tug of gravity on mass m2 would stretch the spring and move m1 at the same time, but I guess we live in a world full of friction so that isn't really accurate.

 

[PJC01]

I would approach this problem by breaking it down into smaller, more manageable parts. First, I would draw a diagram to better visualize the setup and identify all the relevant variables.

Next, I would start by considering the motion of the masses and the pulley separately. For the mass m1, I would use Newton's second law (F = ma) and the spring force equation (F = -kx) to set up a differential equation for its motion. I would also consider the conservation of energy to determine the initial velocity of m1 when m2 is released.

For the pulley, I would use the kinematic equations for rotational motion (such as torque = moment of inertia x angular acceleration) to determine its angular acceleration and the tension in the string.

Once I have equations for the individual components, I would combine them to solve for the motion of m1, taking into account the constraints of the pulley and the length of the spring. I would also consider the conservation of energy for the entire system to check my results.

In terms of finding the initial length of the spring, I would start by considering the equilibrium length of the spring when m2 is released, and then use the spring force equation to determine the initial extension of the spring. This would also be confirmed by the conservation of energy for the system.

Overall, this problem requires a combination of kinematics, dynamics, and energy conservation principles, and it may require some trial and error to find the correct approach. I would also recommend breaking it down into smaller parts and solving them separately before combining them into a final solution.