Spring and Rock Homework: Finding Separation

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SUMMARY

The discussion focuses on determining the separation point between a rock and a spring in a simple harmonic motion scenario. The key equations involved are the potential energy equation U = (1/2)kx^2 and the period equation T = 2 * π * (m/k)^(1/2). The separation occurs when the acceleration of the spring becomes less than the acceleration due to gravity (g). The participants clarify that the rock will separate from the spring when the spring's upward acceleration is insufficient to counteract gravity's downward force.

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  • Understanding of simple harmonic motion principles
  • Familiarity with Newton's second law of motion
  • Knowledge of potential energy in spring systems
  • Basic grasp of oscillation frequency and angular frequency
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Homework Statement



tnrkrkS7.jpg


Homework Equations



U = (1/2)kx^2
T = 2 * π * (m/k)^ (1/2)

The Attempt at a Solution


[/B]
T = 2 * π * (m/k)^ (1/2)
T ^ 2 = 4 * π^2 * (m/k)
k = (4 * π^2 * m) / T^2

So I know the initial potential energy is:

U = (1/2)kx^2
U = (1/2)x^2 * ((4 * π^2 * m) / T^2)

However, I don't know how to find where the rock and the spring separate?
 

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Fontseeker said:

Homework Statement



View attachment 225095

Homework Equations



U = (1/2)kx^2
T = 2 * π * (m/k)^ (1/2)

The Attempt at a Solution


[/B]
T = 2 * π * (m/k)^ (1/2)
T ^ 2 = 4 * π^2 * (m/k)
k = (4 * π^2 * m) / T^2

So I know the initial potential energy is:

U = (1/2)kx^2
U = (1/2)x^2 * ((4 * π^2 * m) / T^2)

However, I don't know how to find where the rock and the spring separate?
There is a constant force of gravity acting on the rock. When happens when the acceleration of gravity is greater than the acceleration provided by the spring? What happens when it is less?
 
tnich said:
There is a constant force of gravity acting on the rock. When happens when the acceleration of gravity is greater than the acceleration provided by the spring? What happens when it is less?
When the acceleration of gravity is greater than the spring, then they separate right?
 
Fontseeker said:
When the acceleration of gravity is greater than the spring, then they separate right?
You are close, but that's not quite it. Will the rock separate when it is traveling downward, or upward?
 
tnich said:
You are close, but that's not quite it. Will the rock separate when it is traveling downward, or upward?
The rock will separate as it is traveling upward, as soon as the spring stretches out (reaches its max amplitude), right?
 
Fontseeker said:
The rock will separate as it is traveling upward, as soon as the spring stretches out (reaches its max amplitude), right?
Nope. Your were closer before. The separation point has to do with the acceleration.
 
tnich said:
Nope. Your were closer before. The separation point has to do with the acceleration.
The spring will be acceleration up, and as long as the acceleration of the spring is greater than g, then so will the rock. As soon as the acceleration of the spring is less than g, it will separate right?
 
Fontseeker said:
The spring will be acceleration up, and as long as the acceleration of the spring is greater than g, then so will the rock. As soon as the acceleration of the spring is less than g, it will separate right?
Right in a sense. You need to think about the signs of the accelerations.
 
tnich said:
Right.
Now, I would just have to find the x at which the accelerations of both the spring and the rock equal the same using Newton's second law right?
 
  • #10
Fontseeker said:
Now, I would just have to find the x at which the accelerations of both the spring and the rock equal the same using Newton's second law right?
You were quick. I edited my answer. You still don't have it quite right. When the spring is accelerating upward and gravity is accelerating downward, the rock stay in contact. When the spring starts accelerating downward and it's acceleration downward is greater than the acceleration of gravity, then the rock will separate.
The question wants to know the amplitude at which the rock will separate. What equations do you know about simple harmonic motion that have acceleration and amplitude in them?
 
  • #11
tnich said:
You were quick. I edited my answer. You still don't have it quite right. When the spring is accelerating upward and gravity is accelerating downward, the rock stay in contact. When the spring starts accelerating downward and it's acceleration downward is greater than the acceleration of gravity, then the rock will separate.
The question wants to know the amplitude at which the rock will separate. What equations do you know about simple harmonic motion that have acceleration and amplitude in them?
I know a(t) = -Aw^2 * sin(wt). Now, I am trying to find when the acceleration starts accelerating downward, so I can just set to 0 right?
 
  • #12
Fontseeker said:
I know a(t) = -Aw^2 * sin(wt). Now, I am trying to find when the acceleration starts accelerating downward, so I can just set to 0 right?
If you were holding a rock stationary on the palm of your hand, how fast would you have to accelerate your hand downward so that rock would fall freely?
 
  • #13
jbriggs444 said:
If you were holding a rock stationary in your hand, how fast would you have to accelerate your hand downward so that rock would fall freely?
I would have to accelerate it down at g.
 
  • #14
Fontseeker said:
I would have to accelerate it down at g.
Right! So how fast does the block on which the fragment is resting have to accelerate downward so that the fragment falls free?
 
  • #15
jbriggs444 said:
Right! So how fast does the block on which the fragment is resting have to accelerate downward so that the fragment falls free?
I am not really understanding your question
 
  • #16
Fontseeker said:
I am not really understanding your question
Replace your palm with a block. How fast does the block have to accelerate downward so that a rock sitting on top falls free?
 
  • #17
jbriggs444 said:
Replace your palm with a block. How fast does the block have to accelerate downward so that a rock sitting on top falls free?
It has to accelerate down at 9.8m/s^2
 
  • #18
Fontseeker said:
It has to accelerate down at 9.8m/s^2
Right! Now all you have to do is figure out what amplitude of an oscillation results in the block having a downward 9.8 m/s^2 acceleration at some point.
 
  • #19
jbriggs444 said:
Right! Now all you have to do is figure out what amplitude of an oscillation results in the block having a downward 9.8 m/s^2 acceleration at some point.

I could use this equation, but I don't know w: a(t) = -Aw^2 * sin(wt)
 
  • #20
Fontseeker said:
I could use this equation, but I don't know w: a(t) = -Aw^2 * sin(wt)
You know that the maximum downward acceleration occurs when sin(ωt) = what?

Edit: and you do know ω -- it is a function of f.
 
  • #21
jbriggs444 said:
You know that the maximum downward acceleration occurs when sin(ωt) = what?

Edit: and you do know ω -- it is a function of f.
sin(wt) = g / (Aw^2) right?

Would you find w using conservation of energy or how would you find it?
 
  • #22
Write omega in terms of f. The multiple choice answers you are given use f.

Edit: Also, answer the question: For a given amplitude A, the maximum downward acceleration of the block occurs when sin(ωt) = what?
 
  • #23
jbriggs444 said:
Write omega in terms of f. The multiple choice answers you are given use f.

Edit: Also, answer the question: For a given amplitude A, the maximum downward acceleration of the block occurs when sin(ωt) = what?

sin(wt) = g / (Aw^2) and then replace w using frequency right?
 
  • #24
Fontseeker said:
sin(wt) = g / (Aw^2) and then replace w using frequency right?
Way easier than that.

The acceleration is maximum when sin(ωt) is maximum. What is the maximum value of sin(ωt)?
 

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