Simple Harmonic Motion with 2 Springs attached to a Dynamics Cart

In summary, the conversation discusses a lab report where the speaker is trying to derive formulas for the Theories and Derivations section. They explain the set-up of the experiment, where two springs are attached to a dynamics cart and a bumper on each end of a track. The cart is pulled and released, and the time it takes to complete five periods is recorded. The speaker knows how to calculate the spring constant for the system, but is having trouble proving theoretically that the spring constants of each spring should add together. Another user explains that the net force on the cart is the sum of the two spring forces, and at equilibrium, the net force is zero. They also provide a derivation for x = Acos(ωt + φ) that
  • #1
DGalt
4
0
I'm trying to write a lab report but I cannot figure out how to derive the formulas for my Theories and Derivations section.
Basically, the set-up was this:
Two springs were attached to either ends of a dynamics cart. Each spring was then attached to a bumper at either end of the dynamics track (basically just something to attach the springs to). The cart was then pulled to one end of the track (compressing one spring and stretching the other) and then released. The time it took the cart to complete five periods was recorded, and we needed to calculate the spring constant for the system from this.

Now, I know how to do the math from the actual lab data. That's not the problem. The spring constants should be additive (they kinda were for our experiment, but we had some pretty bad percent error). Using the equation T = 2(Pi)Sqrt(m/k) we solved for the k of the system.

Now, my problem is proving theoretically that the spring constants of each spring should add to produce the spring constant for the whole system.

The reasoning I've gone through so far is this:
If there were no outside forces, the system would oscillate forever with a constant x for each respective spring. However, since the spring constants for the two springs were different, these x values will be different for each spring. I know that while the acceleration is increasing for one spring, it is decreasing for another. Overall, though, the force should be constant (I think) because the only force that's being applied to the system is that of the moving dynamics cart.

The problem that I'm having is proving that the restoring force is F= (k1+k2)x. In my head I know it is, but my TA is really picky when it comes to our Theories and Derivations and I can't seem to actually figure out what equations will lead me to this final relationship.

I hope I've been clear in what I need help with. Also, if this post belongs in another part of the forum let me know and I'll move it. It just didn't seem to belong in the actual homework section since, well, it's not really a homework problem.

Thanks in advance
 
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  • #2
DGalt said:
The problem that I'm having is proving that the restoring force is F= (k1+k2)x. In my head I know it is, but my TA is really picky when it comes to our Theories and Derivations and I can't seem to actually figure out what equations will lead me to this final relationship.
The net force on the cart is the sum of the two spring forces. At the equilibrium position, the net force is zero. If you move the cart a distance of x to the right, the left spring pulls with an additional force of -k1x (minus just means to the left) and the right spring pushes with an additional force of -k2x. The total restoring force is thus F = -(k1 + k2)x.

Make sense?
 
  • #3
Thanks, that makes sense. I just had to go through the derivation for x = Acos(ωt + φ), which ended up after a bunch of work showing (k1 + k2) / m = ω2 , which is what he wanted
 

1. What is simple harmonic motion?

Simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement of the object from its equilibrium position. This creates a sinusoidal or wave-like motion.

2. How do two springs attached to a dynamics cart affect simple harmonic motion?

When two springs are attached to a dynamics cart, it creates a system with two degrees of freedom. This means that the cart can move in two directions, creating a more complex type of simple harmonic motion known as double or coupled oscillation.

3. How can the spring constant affect the motion of the cart in this system?

The spring constant, or stiffness, of the springs determines how much force is required to stretch or compress the springs. A higher spring constant will result in a higher frequency and shorter period of oscillation, while a lower spring constant will result in a lower frequency and longer period.

4. What factors can affect the amplitude of the motion in this system?

The amplitude of the motion in this system can be affected by the initial displacement, the mass of the cart, the spring constants, and the damping force. Increasing any of these factors can result in an increase in amplitude.

5. How does the damping force affect the motion of the cart in this system?

The damping force, or the resistance to motion, can cause the amplitude of the motion to decrease over time. This is known as damped oscillation. The amount of damping can be controlled by adjusting the friction or air resistance in the system.

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