Amplitude vs Location when KE=PE?

[bob tran]

Homework Statement

A 0.50-kg object is attached to an ideal massless spring of spring constant 20 N/m along a horizontal, frictionless surface. The object oscillates in simple harmonic motion and has a speed of 1.5 m/s at the equilibrium position.
(a) What is the amplitude of vibration?
(b) At what location are the kinetic energy and the potential energy of the system the same?
Correct Answer: a) 0.24 m, b) 0.17 m
I'm not sure how part A is different from part B.

Homework Equations

Part A:
$$\dfrac{1}{2}mv^2=\dfrac{1}{2}kA^2\\ \dfrac{1}{2}mv^2=\dfrac{1}{2}kx^2$$

The Attempt at a Solution

Part A:
$$\dfrac{1}{2}mv^2=\dfrac{1}{2}kA^2\\ A=\sqrt{\dfrac{mv^2}{k}}\\ A=\sqrt{\dfrac{(.5)(1.5)^2}{20}}=0.24 \ \texttt{m}$$
Part B:
Same thing..?
$$\dfrac{1}{2}mv^2=\dfrac{1}{2}kx^2\\ x=\sqrt{\dfrac{mv^2}{k}}\\ x=\sqrt{\dfrac{(.5)(1.5)^2}{20}}=0.24 \ \texttt{m}$$

 

[JeremyG]

x = 0.24m being the amplitude means that the kinetic energy at that point is clearly zero. For part B, remember that the total energy remains the same.

 

[bob tran]

x = 0.24m being the amplitude means that the kinetic energy at that point is clearly zero. For part B, remember that the total energy remains the same.

Hmm.. I still don't understand how this would get 0.17 m as an answer for part B.

 

[JeremyG]

If the total energy remains the same throughout the motion, at the point where KE=PE, what is the expression of the PE/KE in terms of the total energy E?

 

[bob tran]

If the total energy remains the same throughout the motion, at the point where KE=PE, what is the expression of the PE/KE in terms of the total energy E?

$$\dfrac{PE}{KE}=1 \ ?\\ E = 2PE = 2KE \ ?$$

 

[JeremyG]

Yes, and you already know the total energy, correct? Substitute in the expression for PE and you'll be able to solve for the position.

 

[bob tran]

Yes, and you already know the total energy, correct? Substitute in the expression for PE and you'll be able to solve for the position.

Isn't it still 0.24 m?
$$E = PE + KE\\ E = \dfrac{1}{2}kx^2+\dfrac{1}{2}mv^2\\ 2PE = \dfrac{1}{2}kx^2+\dfrac{1}{2}mv^2\\ 2(\dfrac{1}{2}kA^2)= \dfrac{1}{2}kx^2+\dfrac{1}{2}mv^2\\ kA^2= \dfrac{1}{2}kx^2+\dfrac{1}{2}mv^2\\ kA^2-\dfrac{1}{2}mv^2= \dfrac{1}{2}kx^2\\ \sqrt{\dfrac{2(kA^2-\dfrac{1}{2}mv^2)}{k}}=x\\ \sqrt{\dfrac{2((2)(0.24)^2-\dfrac{1}{2}(.5)(1.5)^2)}{20}}=x\\ 0.24 \ \texttt{m} = x$$

 

[JeremyG]

No, look through your steps once again. In particular, what does your 4th step say? That the total energy is 2 times of 1/2kA²? How is that possible? The concept being tested here is once again, that the total energy is conserved throughout the motion.

 

[JeremyG]

Another hint for you though. Instead of thinking in terms of E = 2PE = 2KE, why not think of it in terms of KE = PE = 1/2E?

Mathematically equivalent but I hope this makes it clearer with a shift of perspective

 

[bob tran]

Another hint for you though. Instead of thinking in terms of E = 2PE = 2KE, why not think of it in terms of KE = PE = 1/2E?

Mathematically equivalent but I hope this makes it clearer with a shift of perspective

Still don't see it.
The only way I got an answer that rounds to 0.17 m was by doing:
$$E=PE+KE\\ E=\dfrac{1}{2}mv^2+\dfrac{1}{2}kA^2=1.125 \ \texttt{J}$$
And then using that value here:
$$\dfrac{E}{2}=PE+\dfrac{KE}{2}\\ E=2PE+KE\\ E=2(\dfrac{1}{2}kx^2)+\dfrac{1}{2}mv^2\\ E=kx^2+\dfrac{1}{2}mv^2\\ E-\dfrac{1}{2}mv^2=kx^2\\ \sqrt{\dfrac{E-\dfrac{1}{2}mv^2}{k}}=x\\ \sqrt{\dfrac{1.125-\dfrac{1}{2}(.5)(1.5)^2}{20}}=x\\ 0.167705 \ \texttt{m}=x\\ 0.17 \ \texttt{m} \approx x$$
But, I am not sure if that is correct and why we are allowed to do $\dfrac{E}{2}=PE+\dfrac{KE}{2}\\$, let alone why we should use this step at all to get the correct answer.

 

[lychette]

$$\dfrac{PE}{KE}=1 \ ?\\ E = 2PE = 2KE \ ?$$

I think you need to see that TOTAL energy E =( PE at displacement x) + (KE at displacement x)
When displacement x = 0,...PE =0 and KE is max
When displacement x = A,... KE = 0 and PE is max
so it is correct to say that E = PEmax
or E = KEmax

 

[haruspex]

$$\dfrac{E}{2}=PE+\dfrac{KE}{2}$$

Where did that come from?
You know the total energy. How much PE is there when there's equal KE and PE? What extension does that give?

 

[bob tran]

I think you need to see that TOTAL energy E =( PE at displacement x) + (KE at displacement x)
When displacement x = 0,...PE =0 and KE is max
When displacement x = A,... KE = 0 and PE is max
so it is correct to say that E = PEmax
or E = KEmax

Still not sure how I use this for the problem.

Where did that come from?
You know the total energy. How much PE is there when there's equal KE and PE? What extension does that give?

$\dfrac{E}{2}=PE$? But, what exactly am I supposed to do with that? When I substitute this into the equation, it still get 0.24 m as the answer. It seems like I don't understand something fundamental about this or I keep substituting in the wrong thing.

 

[haruspex]

$\dfrac{E}{2}=PE$? But, what exactly am I supposed to do with that? When I substitute this into the equation, it still get 0.24 m as the answer. It seems like I don't understand something fundamental about this or I keep substituting in the wrong thing.

I see nowhere in this thread where you have done that correctly.
In your post #7, there is an error going from your second line to the third. This path was not going to work anyway because at the end you still have a v in there, and you do not know what v is at the point of interest.
Please try using $\dfrac{E}{2}=PE$ and post your steps.

 

[bob tran]

$$k=20, m=0.5, v=1.5, x=?\\ E=PE+KE\\ E=\dfrac{E}{2}+KE\\ E-\dfrac{E}{2}=KE\\ \dfrac{E}{2}=KE\\ \dfrac{E}{2}=\dfrac{1}{2}mv^2\\ E=mv^2\\ \dfrac{1}{2}kx^2+\dfrac{1}{2}mv^2=mv^2\\ \dfrac{1}{2}kx^2=\dfrac{1}{2}mv^2\\ ...\\ \texttt{which leads to}\\ x=0.24 \ \texttt{m}$$
Where am I going wrong?
Or if I just use $\dfrac{E}{2}=PE$ itself:
$$\dfrac{E}{2}=PE\\ \dfrac{1.125}{2}=\dfrac{1}{2}kx^2\\ 1.125=kx^2\\ \sqrt{\dfrac{1.125}{k}}=x\\ \sqrt{\dfrac{1.125}{20}}=x\\ 0.24 \ \texttt{m}=x$$
Same answer..

 

[nrqed]

$$k=20, m=0.5, v=1.5, x=?\\ E=PE+KE\\ E=\dfrac{E}{2}+KE\\ E-\dfrac{E}{2}=KE\\ \dfrac{E}{2}=KE\\ \dfrac{E}{2}=\dfrac{1}{2}mv^2\\ E=mv^2\\ \dfrac{1}{2}kx^2+\dfrac{1}{2}mv^2=mv^2\\ \dfrac{1}{2}kx^2=\dfrac{1}{2}mv^2\\ ...\\ \texttt{which leads to}\\ x=2.4 \ \texttt{m}$$
Where am I going wrong?

All your algebra steps are correct. It is only when you plug in the values and solve for x that you somehow make a mistake. You should be getting 0.24 m, for some reason you are off by a factor of 10. Did you use the values given in your first post?
(By the way, there are actually two possible answers for x, 0.24 m and -0.24 m. I don't know why they choose to give only the positive answer)

 

[bob tran]

All your algebra steps are correct. It is only when you plug in the values and solve for x that you somehow make a mistake. You should be getting 0.24 m, for some reason you are off by a factor of 10. Did you use the values given in your first post?
(By the way, there are actually two possible answers for x, 0.24 m and -0.24 m. I don't know why they choose to give only the positive answer)

Sorry, it is 0.24 m. I just accidentally typed 2.4 m (fixed it in previous post now).

But, I am trying to get 0.17 m for part B, not 0.24 m.

 

[lychette]

$$k=20, m=0.5, v=1.5, x=?\\ E=PE+KE\\ E=\dfrac{E}{2}+KE\\ E-\dfrac{E}{2}=KE\\ \dfrac{E}{2}=KE\\ \dfrac{E}{2}=\dfrac{1}{2}mv^2\\ E=mv^2\\ \dfrac{1}{2}kx^2+\dfrac{1}{2}mv^2=mv^2\\ \dfrac{1}{2}kx^2=\dfrac{1}{2}mv^2\\ ...\\ \texttt{which leads to}\\ x=0.24 \ \texttt{m}$$
Where am I going wrong?
Or if I just use $\dfrac{E}{2}=PE$ itself:
$$\dfrac{E}{2}=PE\\ \dfrac{1.125}{2}=\dfrac{1}{2}kx^2\\ 1.125=kx^2\\ \sqrt{\dfrac{1.125}{k}}=x\\ \sqrt{\dfrac{1.125}{20}}=x\\ 0.24 \ \texttt{m}=x$$
Same answer..

$$k=20, m=0.5, v=1.5, x=?\\ E=PE+KE\\ E=\dfrac{E}{2}+KE\\ E-\dfrac{E}{2}=KE\\ \dfrac{E}{2}=KE\\ \dfrac{E}{2}=\dfrac{1}{2}mv^2\\ E=mv^2\\ \dfrac{1}{2}kx^2+\dfrac{1}{2}mv^2=mv^2\\ \dfrac{1}{2}kx^2=\dfrac{1}{2}mv^2\\ ...\\ \texttt{which leads to}\\ x=0.24 \ \texttt{m}$$
Where am I going wrong?
Or if I just use $\dfrac{E}{2}=PE$ itself:
$$\dfrac{E}{2}=PE\\ \dfrac{1.125}{2}=\dfrac{1}{2}kx^2\\ 1.125=kx^2\\ \sqrt{\dfrac{1.125}{k}}=x\\ \sqrt{\dfrac{1.125}{20}}=x\\ 0.24 \ \texttt{m}=x$$
Same answer..

Still not sure how I use this for the problem.

$\dfrac{E}{2}=PE$? But, what exactly am I supposed to do with that? When I substitute this into the equation, it still get 0.24 m as the answer. It seems like I don't understand something fundamental about this or I keep substituting in the wrong thing.

Sorry, it is 0.24 m. I just accidentally typed 2.4 m (fixed it in previous post now).

But, I am trying to get 0.17 m for part B, not 0.24 m.

In your first post you used max KE = max PE to calculate the amplitude...great...you are more than 1/2 way there
Max PE = 1/2kA^2... PE at any other value of displacement, x is = 1/2kx^2
What value of x gives you 1/2 the max energy?...

 

[nrqed]

All your algebra steps are correct. It is only when you plug in the values and solve for x that you somehow make a mistake. You should be getting 0.24 m, for some reason you are off by a factor of 10. Did you use the values given in your first post?
(By the way, there are actually two possible answers for x, 0.24 m and -0.24 m. I don't know why they choose to give only the positive answer)

Ah, sorry I had misread the first post.

I see. The problem here is that in your solution, the v should be the speed when the object has PE = KE, but this speed is not 1.5 m/s (that's the maximum speed). Yours steps are correct, but using v=1/5 m/s in the final step is the mistake. When writing KE and PE, on must keep track where these are evaluated.

What you do know is that E = KE + PE and that at the equilibrium position, where PE =0, the speed is 1.5 m/s You therefore have a value for the total energy.

When you write E/2 = KE, this is correct but keep in mind that on the right side, the speed in 1/2 mv^2 is the speed at the position where KE=PE so this is an unknown. You can use that equation to find the speed at that point if you want, although it is not asked so you don't have to.

It is better to write E = PE + KE as E = PE + E/2 at the point of interest, and then solve for x.

 

[bob tran]

Ah, sorry I had misread the first post.

I see. The problem here is that in your solution, the v should be the speed when the object has PE = KE, but this speed is not 1.5 m/s (that's the maximum speed). Yours steps are correct, but using v=1/5 m/s in the final step is the mistake. When writing KE and PE, on must keep track where these are evaluated.

What you do know is that E = KE + PE and that at the equilibrium position, where PE =0, the speed is 1.5 m/s You therefore have a value for the total energy.

When you write E/2 = KE, this is correct but keep in mind that on the right side, the speed in 1/2 mv^2 is the speed at the position where KE=PE so this is an unknown. You can use that equation to find the speed at that point if you want, although it is not asked so you don't have to.

It is better to write E = PE + KE as E = PE + E/2 at the point of interest, and then solve for x.

Okay, hopefully, I got it this time.
$$\text{At the equilibrium position, PE=0, so:}\\ E=\dfrac{1}{2}mv^2=\dfrac{1}{2}(.5)(1.5)^2=.5625 \text{ J}\\ \text{So, because } \dfrac{E}{2}=KE,\\ E=PE+\dfrac{E}{2}\\ .5625=\dfrac{1}{2}kx^2+\dfrac{.5625}{2}\\ \sqrt{\dfrac{.5625}{k}}=x\\ \sqrt{\dfrac{.5625}{20}}=x\\ .167750983=x\\ 0.17 \text{ m}\approx x$$
Is this the correct way of doing part B?

 

[nrqed]

Okay, hopefully, I got it this time.
$$\text{At } PE=KE,\\ E=\dfrac{1}{2}mv^2=\dfrac{1}{2}(.5)(1.5)^2=.5625 \text{ J}\\ \text{So, because } \dfrac{E}{2}=KE,\\ E=PE+\dfrac{E}{2}\\ .5625=\dfrac{1}{2}kx^2+\dfrac{.5625}{2}\\ \sqrt{\dfrac{.28125}{k}}=x\\ \sqrt{\dfrac{.28125}{20}}=x\\ .167750983=x\\ 0.17 \text{ m}\approx x$$
Is this the correct way of doing part B?

Yes, that's perfect.
What is important is to keep track (if not on paper at least in your head) of where the KE and PE are evaluated whenever you write a relation. For example, when you write E/2=KE , you have to remember that this true only at the point where KE = PE.

But yes, your derivation is correct.

 

[haruspex]

$$k=20, m=0.5, v=1.5, x=?\\ E=PE+KE\\ E=\dfrac{E}{2}+KE\\ E-\dfrac{E}{2}=KE\\ \dfrac{E}{2}=KE\\ \dfrac{E}{2}=\dfrac{1}{2}mv^2\\ E=mv^2\\ \dfrac{1}{2}kx^2+\dfrac{1}{2}mv^2=mv^2\\ \dfrac{1}{2}kx^2=\dfrac{1}{2}mv^2\\ ...\\ \texttt{which leads to}\\ x=0.24 \ \texttt{m}$$
Where am I going wrong?
Or if I just use $\dfrac{E}{2}=PE$ itself:
$$\dfrac{E}{2}=PE\\ \dfrac{1.125}{2}=\dfrac{1}{2}kx^2\\ 1.125=kx^2\\ \sqrt{\dfrac{1.125}{k}}=x\\ \sqrt{\dfrac{1.125}{20}}=x\\ 0.24 \ \texttt{m}=x$$
Same answer..

E is not 1.125J. How did you get that? It should be half that.

 

[lychette]

In your first post you used max KE = max PE to calculate the amplitude...great...you are more than 1/2 way there
Max PE = 1/2kA^2... PE at any other value of displacement, x is = 1/2kx^2
What value of x gives you 1/2 the max energy?...

In your first post you used max KE = max PE to calculate the amplitude...great...you are more than 1/2 way there
Max PE = 1/2kA^2... PE at any other value of displacement, x is = 1/2kx^2
What value of x gives you 1/2 the max energy?...[

Thank you...you had the answer in reach in your first post...you were diverted from the straight and narrow and got lost in the woods :)