# Unequal mass spring system

• FunkyDwarf
In summary, the center of mass does not move, so imagine two smaller springs are fixed there and each has a mass at the end. Pick one of those springs. What's its spring constant? What's its mass? What's its frequency? It's just a mass on the end of a spring.

#### FunkyDwarf

Hey guys

Im revising for my end of year exams and this was a question i got wrong in a test we had this year and unforunately there arent worked answers. I sort of get close but not quite there.

The answer is A, which i get, sorta
$$F = ma = 2kx$$
As per Newtons third law (the hint) saying that the pulling and pushing force in turn create a pushing and pulling force, thus doubling the force. So using
$$s = ut +1/2 at^2$$
Now the different masses will go different distances so if we take a ratio of masses, ie the smaller one going further we get
$$\frac{4x}{5} = \frac{kxt^2}{m}$$
$$\frac{4}{5} = \frac{kt^2}{m}$$

here I am trying to use t as period and thus get f via T = 1/f
Now this works fine, you rearange for t and you get the answer A. However one thing that puzzels me is: if this is supposed to be the period, surely we should consider one whole oscilation, that is to say the distance would infact be 8/5, which throws a spanner in the works. Any ideas?

Cheers
-G

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Hint: What point doesn't move at all throughout the oscillation?

well the centre of mass, and i realize now I am missing something obvious, but how does that help me?

CM = 4/5L or x+L depending on length (from the small mass) right? and 1/5 away from the big one? please help I am willing to look stupid and learn if i can get to the bottom of this, itsb een bugging me all day :P

FunkyDwarf said:
well the centre of mass, and i realize now I am missing something obvious, but how does that help me?

CM = 4/5L or x+L depending on length (from the small mass) right? and 1/5 away from the big one? please help I am willing to look stupid and learn if i can get to the bottom of this, itsb een bugging me all day :P
I assume you know how the force on each mass depends on the Δx of the spring. Imagine adding a rigid clamp to the spring at that point that does not move. That would have no effect on the motion on either side of the clamp. You could cut one side of the spring off, and the other side with its mass would continue to oscillate with the same frequency it had before the clamp was added. What does cutting a spring do to its behavior?

Yeh but that's what I've done isn't it? I am asking if my stuff is right, obviously your posting because its not and i would like to know which part. whilst i appreciate the riddle aspect and i don't get something for nothing, I've presented working here. I mean i get the right answer, but there's the discrepincy about the distance traveled still.

Yes, you did show some work, but I don't understand it. Something about Newton's III implying a doubling of the force? Better to start over with the hints we gave.

The center of mass does not move. So pretend two smaller springs are fixed at that point and each has a mass at the end. Pick one of those springs. What's its spring constant? What's its mass? What's its frequency? It's just a mass on the end of a spring. (I assume you know how to find the frequency of oscillation of a single mass on a spring.)

Well no offence but my work isn't that hard to follow. I understand your point . Youre saying the force is simply kx where x is a different distance for each mass, as i worked out before. I tried this way but it leads to the wrong answer.

I used s = ut + 1/2at^2 with ut = 0 as initial velocity is zero. Then if F = ma = kx then 1/5 (for large mass) = (1/2)(k/4m)t^2 which gives root 8/5 not root 4/5. Furthermore the point which i raised as my problem but which hasnt been addressed is should s be x or 2x? Ie the distance traveled in one period.

Whilst your hints are helpful id really like some serious numerical help here as I've done (initially anyway) what you guys suggested but I am getting nowhere.

FunkyDwarf said:
Well no offence but my work isn't that hard to follow. I understand your point . Youre saying the force is simply kx where x is a different distance for each mass, as i worked out before. I tried this way but it leads to the wrong answer.
The spring exerts the same force on each mass, that's for sure. And each mass does move a different distance as it oscillates. But the center of mass of the system is fixed.

I'm saying that an easy way to solve this problem is to treat each mass as being on its own, separate (shorter) spring, with a different spring constant than the original. Did you try that?

I used s = ut + 1/2at^2 with ut = 0 as initial velocity is zero. Then if F = ma = kx then 1/5 (for large mass) = (1/2)(k/4m)t^2 which gives root 8/5 not root 4/5. Furthermore the point which i raised as my problem but which hasnt been addressed is should s be x or 2x? Ie the distance traveled in one period.
Why are you using a kinematics formula for constant acceleration? It's hard to offer a comment when I don't see what you are trying to do.

Whilst your hints are helpful id really like some serious numerical help here as I've done (initially anyway) what you guys suggested but I am getting nowhere.
Why don't you try following our suggestions? (Once you see what we're saying, you'll be done in 60 seconds.) Treat the spring as being in two pieces, connected at the center of mass. Answer these questions:
(1) How long is each spring?
(2) What's the spring constant of each spring?
(3) What's the frequency of oscillation of each mass on its spring?

The reason I am using that forumula is because it contains constants i have and the variable i need. I know the distance it moves and the accleration it feels and the time variable is there. Also its not constant acceleration, the force and therefore accel changes with distance.

1.) Well its all in variables so all i can say is the distance from the equilibrium position each one moves, which is all that really matters as if you look in the possible answers there's no L, plus its logical.
2.) this doesn't come into it given the question so we just use k
3.) Again, using that formula you can get t for one period which is T and then 1/T is f.

What is wrong with this? I don't get what youre saying

FunkyDwarf said:
The reason I am using that forumula is because it contains constants i have and the variable i need. I know the distance it moves and the accleration it feels and the time variable is there. Also its not constant acceleration, the force and therefore accel changes with distance.

1.) Well its all in variables so all i can say is the distance from the equilibrium position each one moves, which is all that really matters as if you look in the possible answers there's no L, plus its logical.
2.) this doesn't come into it given the question so we just use k
3.) Again, using that formula you can get t for one period which is T and then 1/T is f.

What is wrong with this? I don't get what youre saying
When I first replied, you had already said you recognized that you were missing something. I will be more specific about what you are missing, and in doing so I will repeat some of what Doc Al has been saying. I will add some coments to your original post to put those comments into context.

FunkyDwarf said:
Hey guys

Im revising for my end of year exams and this was a question i got wrong in a test we had this year and unforunately there arent worked answers. I sort of get close but not quite there.

The answer is A, which i get, sorta
$$F = ma = 2kx$$
As per Newtons third law (the hint) saying that the pulling and pushing force in turn create a pushing and pulling force, thus doubling the force.

This equation is not correct. The observation is not correct. The hint about Newton's 3rd law is not about pushing vs pulling. There is no doubling of the force. By Newton's 3rd law there are two equal and opposite forces, an action-reaction pair, at each end of the spring. Each mass exerts a force on the spring and each mass experiences an equal and opposite force from the spring.

It takes forces applied in opposite directions at opposite ends of the spring to either stretch or compress the spring. When those two forces acting on the spring (not an action reaction pair- they are acting on the same object) are outward it stretches, and when they are acting inward it compresses. For simplicity of notation and because you used just x I will use x (instead of the Δx in the problem statement) to represent the elongation of the sring. The amount by which the length of the spring changes is related to the magnitude of this force by

F = kx

When x is positive, the spring is stretched and when x is negative the spring is compressed. By Newton's 3rd law, a force of this magnitude is experienced by each mass. There is no F = 2kx

So using
$$s = ut +1/2 at^2$$

This equation is not applicable to any situation where the force and acceleration are not constant, which is the case in this problem.

Now the different masses will go different distances

This is true. They will go different distances, and it appears you have figured out the correct ratio of the distances they move.

so if we take a ratio of masses, ie the smaller one going further we get
$$\frac{4x}{5} = \frac{kxt^2}{m}$$
$$\frac{4}{5} = \frac{kt^2}{m}$$
here I am trying to use t as period and thus get f via T = 1/f

I see no connection between this and anything you have written previously. The justification for introducing a factor of 4/5 or 5/4 into the problem is not established by writing the first of these equations, even if it does happen to reduce the "right answer." There is no relationship between the period of an oscillator and the amount by which the spring is stretched or compressed at any time during its motion.

Now this works fine, you rearange for t and you get the answer A. However one thing that puzzels me is: if this is supposed to be the period, surely we should consider one whole oscilation, that is to say the distance would infact be 8/5, which throws a spanner in the works. Any ideas?

I have no idea what you mean by the distance being 8/5, but your comment shows that you do not really know why the 5/4 factor enters into the answer to the problem.

Cheers
-G
The thing we have been trying to lead you to is recognizing that each mass experiences an equal and opposite force of magnitude

F = kx

For each mass this force tends to restore the mass to its equilibrium position. However, the x in this equation is not the displacement of the mass. To apply the formulas for a harmonic oscillator, you need to know the relationship between the force applied to a moving mass and the distance that mass moves. You recognize that one mass moves more than the other, and it is obvious from what you have done that you can write the distance each mass moves in terms of x. Your next step should be to write two equations, one for each mass, that expresses the force acting on that mass in terms of the distance that mass moves. Once you have those equations, you will have an effective spring constant (not k, but proportional to k) for each mass and then you can apply the usual equations for harmonic motion. Either of the two equations will lead you to the same conclusion about the frequency of oscillation shared by the two masses.

Thats better :) As for the last blue bit i do know why its just i didnt express it very well.

As for the rest of it I am a bit stuck. We never actually did this sort of thing in classes, its a general mechanics course (well not course but sub unit) and we just have the general equations of motion to work with.

That being said i can see how you can derive the situation for just one m where f = 2pi sqrt(k/m) that's fine i get the derivation of that. But when i try an incorporate the variable distances traveled by each mass i get the displacement canceling out which is obviously useless (and probably wrong)

if (ignoring L) $$x = Acos(wt)$$ and say its the big mass so its
$$x = (1/5)Cos(wt)$$ then $$a = -w^2Cos(wt) = -w^2 x$$ as per the rules and definition of SHM (this second x of course being representative of the cos related function)However, any way i try it i still just get the basic form, instead of one involving 5/4.

ie $$F = ma = -k[(1/5)Cos(wt)] = m[-w^2(1/5)Cos(wt)] = -k(1/5)Cos(wt)$$
:S help! i know I am missing something obvious and looking like a moron but that's nothing new for me :P well the 2nd part anyway

FunkyDwarf said:
Thats better :) As for the last blue bit i do know why its just i didnt express it very well.

As for the rest of it I am a bit stuck. We never actually did this sort of thing in classes, its a general mechanics course (well not course but sub unit) and we just have the general equations of motion to work with.

That being said i can see how you can derive the situation for just one m where f = 2pi sqrt(k/m) that's fine i get the derivation of that. But when i try an incorporate the variable distances traveled by each mass i get the displacement canceling out which is obviously useless (and probably wrong)

if (ignoring L) $$x = Acos(wt)$$ and say its the big mass so its
$$x = (1/5)Cos(wt)$$ then $$a = -w^2Cos(wt) = -w^2 x$$ as per the rules and definition of SHM (this second x of course being representative of the cos related function)However, any way i try it i still just get the basic form, instead of one involving 5/4.

ie $$F = ma = -k[(1/5)Cos(wt)] = m[-w^2(1/5)Cos(wt)] = -k(1/5)Cos(wt)$$
:S help! i know I am missing something obvious and looking like a moron but that's nothing new for me :P well the 2nd part anyway
Try using new variables to represent the displacements of the two masses from their equilibrium positions, say y for the big mass and z for the small mass with directions chosen so that both are positive when the spring is stretched (y + z = x). Then write the two equations I suggested you write in the previous post.

F = F(y) = ?
F = F(z) = ?

Assume a sinusoidal solution for y and z as you do for any oscillator and figure out what ω has to be to satisfy each equation.

## 1. What is an unequal mass spring system?

An unequal mass spring system is a physical system that consists of two or more masses connected by springs of different stiffness. The masses are not equal, meaning they have different masses, and the springs have different spring constants.

## 2. How does the unequal mass affect the behavior of the spring system?

The unequal mass affects the behavior of the spring system by changing the natural frequency of the system. The natural frequency is the rate at which the system oscillates when there is no external force acting on it. The unequal mass also affects the amplitude and phase of the oscillations of the system.

## 3. What is the equation of motion for an unequal mass spring system?

The equation of motion for an unequal mass spring system is given by: m1x1'' + k1x1 + k2(x1 - x2) = 0 and m2x2'' + k2(x2 - x1) = 0. Where m1 and m2 are the masses, x1 and x2 are the displacements from equilibrium, and k1 and k2 are the spring constants of the two springs.

## 4. How is the natural frequency of an unequal mass spring system calculated?

The natural frequency of an unequal mass spring system is calculated using the formula: ω = √(k/m), where ω is the natural frequency, k is the spring constant, and m is the mass of the system. For an unequal mass spring system, the natural frequency will be different for each mass.

## 5. What is the significance of unequal mass spring systems in real-life applications?

Unequal mass spring systems have many real-life applications, such as in the suspension of vehicles, shock absorbers, and seismic isolation systems. They are also used in musical instruments, where the unequal masses of the strings or keys create different natural frequencies, producing different tones. In engineering, unequal mass spring systems are used to reduce the effects of vibrations and improve the stability of structures.