Energy conservation and periodic motion

In summary: For small displacements it should be. Otherwise it will not produce simple harmonic oscillation. Presumably what will be needed are the small angle approximations to the sin and cos functions.It looks a lot easier to use energy conservation, as the OP intended.
  • #1
arpon
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Homework Statement


upload_2015-3-2_20-34-40.png

Four weightless rods of length ##l## each are connected by hinged joints and form a rhomb (Fig. 48). A hinge A is fixed, and a load is suspended to a hinge C. Hinges D and B are connected by a weightless spring of length ##1.5l## in the undeformed state. In equilibrium, the rods form angles ##\alpha _0 = 30° ## with the vertical. Determine the period T of small oscillations of the load.

Homework Equations


##U = mgh ##
##U = \frac{1}{2} kx^2 ##

The Attempt at a Solution


I used energy conservation law in this case. But this gave me crazy results. Surely, I have made a mistake to apply this law. Would you please help me to find out the mistake ?
upload_2015-3-2_20-43-2.png

So, ## h = 2l cos \alpha## ... (i) ;
##y = 2l sin \alpha## ;
Expansion (or compression) of the spring, ## x = y - 1.5 l = l ( 2sin \alpha - 1.5)## ...(ii)
Let the spring constant be ##k## and the mass of the load be ##m##;
Applying energy conservation law :
## \frac{1}{2} kx^2 - mgh = constant ## [when h increase, gravitational potential decreases]
## kx \frac {dx}{dt} - mg \frac {dh}{dt} = 0##
##kx - mg \frac {dh}{dx} = 0## ... (iii)
But, from eq. (i) & (ii), ## \frac{dh}{dx} = - tan \alpha ##
So, (iii) >>
## kx = -mg tan \alpha ## ;
## k = - \frac{mg tan \alpha}{x} = - \frac{mg tan \alpha}{ l ( 2sin \alpha - 1.5)} ##
So, ##k## becomes variable.
 
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  • #2
I don't follow your equations. What is x?

I would take x = h - h0 (where h0 is the equilibrium point).
 
  • #3
PeroK said:
I don't follow your equations. What is x?

I would take x = h - h0 (where h0 is the equilibrium point).
x is the expansion of the spring.
 
  • #4
Okay. I missed that. Where's your Kinetic Energy?
 
  • #5
PeroK said:
Okay. I missed that. Where's your Kinetic Energy?
Oh! Sorry! How could I make such a silly mistake!:))
Thanks for your help.
 
  • #6
Oscillation means that neither x nor h are constant. So you don't want the equation you wrote.

Harmonic oscillation comes from a restorative force proportional to the distance away from equilibrium. So you want to try to figure out the force pushing the system back to equilibrium. You have the equilibrium position. Suppose you push the mass up by a short distance, call it d. What will the force be that is required to hold the mass there? It should be some constant times d. And the direction should be such that the system is trying to push the mass back to its equilibrium. That is, the system will provide a force of the following form.

F = - Keff d

Here Keff is the effective spring constant of the system.

Once you have that, then you can do the usual things for harmonic motion to get the period.
 
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  • #7
DEvens said:
Oscillation means that neither x nor h are constant. So you don't want the equation you wrote.

Harmonic oscillation comes from a restorative force proportional to the distance away from equilibrium. So you want to try to figure out the force pushing the system back to equilibrium. You have the equilibrium position. Suppose you push the mass up by a short distance, call it d. What will the force be that is required to hold the mass there? It should be some constant times d. And the direction should be such that the system is trying to push the mass back to its equilibrium. That is, the system will provide a force of the following form.

F = - Keff d

Here Keff is the effective spring constant of the system.

Once you have that, then you can do the usual things for harmonic motion to get the period.

In this case the restorative force is not proportional to the displacement.
 
  • #8
PeroK said:
In this case the restorative force is not proportional to the displacement.

For small displacements it should be. Otherwise it will not produce simple harmonic oscillation. Presumably what will be needed are the small angle approximations to the sin and cos functions.
 
  • #9
It looks a lot easier to use energy conservation, as the OP intended.
 
  • #10
DEvens said:
Oscillation means that neither x nor h are constant. So you don't want the equation you wrote.

Harmonic oscillation comes from a restorative force proportional to the distance away from equilibrium. So you want to try to figure out the force pushing the system back to equilibrium. You have the equilibrium position. Suppose you push the mass up by a short distance, call it d. What will the force be that is required to hold the mass there? It should be some constant times d. And the direction should be such that the system is trying to push the mass back to its equilibrium. That is, the system will provide a force of the following form.

F = - Keff d

Here Keff is the effective spring constant of the system.

Once you have that, then you can do the usual things for harmonic motion to get the period.
I solved this problem using this technique. Thanks for your help.
 
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1. What is energy conservation and why is it important?

Energy conservation is the principle that energy cannot be created or destroyed, but can only be transformed from one form to another. It is important because it helps us understand the ways in which energy is used and how to use it more efficiently, reducing our carbon footprint and helping to combat climate change.

2. How does energy conservation relate to periodic motion?

Periodic motion is a repeating motion that occurs over regular intervals of time. Energy conservation is related to periodic motion because energy is constantly being transferred between different forms during the motion, but the total amount of energy remains the same.

3. What are some examples of energy conservation in everyday life?

Some examples of energy conservation in everyday life include turning off lights and electronics when not in use, using energy-efficient appliances, carpooling or using public transportation, and choosing renewable energy sources.

4. How can we calculate energy conservation in a system?

The law of conservation of energy states that the total energy in a closed system remains constant. To calculate energy conservation in a system, we can use the equation: Energy In = Energy Out. This means that the initial energy in the system must equal the final energy in the system.

5. What are some challenges in implementing energy conservation measures?

Some challenges in implementing energy conservation measures include resistance to change, lack of knowledge or awareness, and upfront costs. However, the long-term benefits, both for the environment and for cost savings, make it essential to overcome these challenges and prioritize energy conservation.

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