Solving SHM of Cube Connected By Rubber Bands

In summary, the cube is displaced by a small distance y perpendicular to the length of the rubber bands, and the system exhibits SHM. Find the angular frequency ω using Vela's equation.
  • #1
NATURE.M
301
0

Homework Statement



A cube of mass m is connected to two rubber bands of length L, each under tension T. The cube is displaced by a small distance y perpendicular to the length of the rubber bands. Assume the tension doesn't change. Show that the system exhibits SHM, and find its angular freqency ω.

The Attempt at a Solution



So basically from a FBD of cube, I have vertical forces: -2Tsinθ - mg = m[itex]\frac{d^2y}{dt^2}[/itex] and the horizontal components of tension from each band cancels. Now since the cube is displaced by a small distance y, I assume we can approximate sinθ ≈ θ. But then I'm not sure what to do?
I tried using sinθ = [itex]\frac{y}{(y^2+L^2)^{1/2}}[/itex], but then I get a complicated expression.
I know I need to obtain a -constant*y on the LHS. Any suggestions.
 
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  • #2
I think i got it, your assuming sinθ ≈ y/L, for small y.
One question though, do we have to assume gravity is negligible to get a sensible answer?
 
Last edited:
  • #3
No, you don't. The differential equation becomes
$$y'' + \frac{2T}{mL}y + g=0.$$ Now consider a change of variables to ##u = y+\frac{mg}{2T}L##. What's the differential equation in terms of ##u##? What does ##\frac{mg}{2T}L## physically represent?
 
  • #4
vela said:
No, you don't. The differential equation becomes
$$y'' + \frac{2T}{mL}y + g=0.$$ Now consider a change of variables to ##u = y+\frac{mg}{2T}L##. What's the differential equation in terms of ##u##? What does ##\frac{mg}{2T}L## physically represent?

We haven't studied differential equations in much depth (since its a introductory physics course), so I didn't really catch the change of variables part. If you could explain further, I'd appreciate it.
 
  • #5
I would just throw gravity out. The problem doesn't specifically mention it, so it might as well be on a frictionless tabletop or so.
 
  • #6
NATURE.M said:
We haven't studied differential equations in much depth (since its a introductory physics course), so I didn't really catch the change of variables part. If you could explain further, I'd appreciate it.
I'm saying rewrite the equation in terms of u instead of y. If you differentiate u twice, you get u''=y'', right? Just substitute in for y and y''.

jackarms said:
I would just throw gravity out. The problem doesn't specifically mention it, so it might as well be on a frictionless tabletop or so.
It would be kind of hard to oscillate vertically on a flat tabletop.
 
  • #7
vela said:
It would be kind of hard to oscillate vertically on a flat tabletop.
When I read the problem, I also assumed (as jackarms did) that the oscillation was taking place in the horizontal, not the vertical. There is nothing in the problem statement that mentions the vertical. I pictured a horizontal frictionless table.

Chet
 
  • #8
this is what I came out with...I also went on to show the frequency as well...
 

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  • #9
Ronaldo95163 said:
this is what I came out with...I also went on to show the frequency as well...
This is definitely not what my answer would have been. I would have had T as a parameter in the frequency. I would have had:
[tex]f=\frac{1}{2π}\sqrt{\frac{2T}{mL}}[/tex]
This is based on Vela's equation in post #3, with g removed.

Chet
 
  • #10
That's true. I inferred it from the use of ##y##, but that's not really justified.
 
  • #11
The angular frequency is actually represented by omega...f is the frequency of oscillation
 
  • #12
Ronaldo95163 said:
The angular frequency is actually represented by omega...f is the frequency of oscillation
Yes. ω=2πf

Chet
 
  • #13
Yip so my proof of the angular frequency stops at the third to last line
 

FAQ: Solving SHM of Cube Connected By Rubber Bands

1. What is SHM?

SHM stands for Simple Harmonic Motion. It is a type of motion in which an object moves back and forth in a regular pattern, such as a swinging pendulum or a vibrating guitar string.

2. How is a cube connected by rubber bands?

A cube connected by rubber bands is a physical model used to study SHM. It consists of a cube made of rigid material, such as wood or plastic, with rubber bands attached to each vertex. The rubber bands act as the restoring force that allows the cube to oscillate back and forth.

3. What factors affect the SHM of a cube connected by rubber bands?

The SHM of a cube connected by rubber bands can be affected by several factors, such as the stiffness of the rubber bands, the mass of the cube, and the amplitude of the oscillations. The length of the rubber bands and the angle at which they are attached to the cube can also impact the SHM.

4. How is the period of oscillation determined for a cube connected by rubber bands?

The period of oscillation for a cube connected by rubber bands is determined by the mass of the cube, the stiffness of the rubber bands, and the length of the rubber bands. The equation for the period is T = 2π√(m/k), where T is the period, m is the mass, and k is the stiffness constant of the rubber bands.

5. What are the practical applications of studying SHM of a cube connected by rubber bands?

Studying SHM of a cube connected by rubber bands can have practical applications in various fields such as engineering, physics, and mathematics. It can help in understanding the behavior of oscillating systems and designing structures that can withstand oscillations. It can also be used to study the properties of materials and develop models for predicting the behavior of more complex systems.

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