Exploring Non-SHM Systems with Energy Equations

[Physicist_FTW]

1. U(x)=U0(x/a)^1000000
Find the period for a mass m, if it has total energy E


2. E=U+K




3. dE/dt=0=v[mdv/dt+dU/dx]

I am really stuck on this one, I am not sure what to do at all talked to my proffessor he says just to re-read the chapter but if I am honest I've always been one to learn through examples which he hasnt given us, any clues would be most appreciated!

 

[berkeman]

1. U(x)=U0(x/a)^1000000
Find the period for a mass m, if it has total energy E


2. E=U+K




3. dE/dt=0=v[mdv/dt+dU/dx]

I am really stuck on this one, I am not sure what to do at all talked to my proffessor he says just to re-read the chapter but if I am honest I've always been one to learn through examples which he hasnt given us, any clues would be most appreciated!

Is this a spring and mass problem, or a pendulum problem or what? Please post more details and the relevant equations in more detail, and show us how you have tried to start the solution...

 

[Physicist_FTW]

its a SHM probelm, well i tried
-dU/dx=F(x)
F(x)=m(d^2x/dt^2)
then i think I am meant to guess a value for x(t) but I am not really sure/

 

[RoyalCat]

its a SHM probelm, well i tried
-dU/dx=F(x)
F(x)=m(d^2x/dt^2)
then i think I am meant to guess a value for x(t) but I am not really sure/

First I suggest that you express the potential energy as:

$$U(x)=\frac{U_0}{a^{k+1}}x^{k+1}$$ where in our case $$k+1=1000000$$

Use the following theorem:

$$F(x)=-U'(x)$$

And from there all that remains is to solve a tricky differential equation. I'm trying it myself, it looks interesting.

 

[Physicist_FTW]

If E=U+K
E=((U0x^K+1)/a^k+1)+0.5(m)(dx/dt)^2
rearrange
dx/dt=(2/m(E-U0x^K+1)/a^k+1))^0.5
is this the right way about, I am not sure how to do this integral.

 

[Gear300]

This system doesn't look to be SHM.