Two masses and spring oscillation

[Tanya Sharma]

Homework Statement

The ratio of the time periods of small oscillation of the insulated spring and mass system before and after charging the masses is

(a) ≥ 1
(b) > 1
(c) ≤ 1
(d) = 1

Homework Equations

The Attempt at a Solution

First I calculated the time period of neutral masses .

Let us consider origin to be at the left of m1 .Take the x-axis to be along the length of the spring and call the left mass be m1 at coordinate x1 and right mass be m2 at coordinate x2 .

x=x2-x1-L

d²x/dt² = d²(x2-x1)/dt²

EOM for mass m1 is m1d²x1/dt² = kx

EOM for mass m2 is m2d²x2/dt² = -kx

After manipulating the above two equations we arrive at d²/dt² + (k/μ)x = 0 ,where μ = m1m2/(m1+m2) .

The time period is 2π√(μ/k) .

Now when the masses are charged ,

EOM for mass m1 is m1d²x1/dt² = kx - k_eq² / (x+L)²

EOM for mass m2 is m2d²x2/dt² = -kx + k_eq² / (x+L)² .

Is this the correct way to approach the problem ? If yes ,how should I proceed .

 

[NascentOxygen]

This is just one of a few dozen multi-choice questions on an exam paper? If that's the case, there is probably not a lot of calculation to be done.

You are not told whether in this test the weights are changed from a smaller mass to a larger, or whether it's vice-versa? So even if the period were observed to change, how could you say whether the corresponding ratio is >1 when it might be <1 ?

So, purely on the absence of information allowing you to decide on any contrary option, it seems you are confined to only one answer.

insulated spring and mass system before and after charging

spelling: probably "isolated" spring, and after "changing"

 

[ehild]

The potential consists of an elastic part and a Coulomb-part. Expand the potential function into Taylor series about the equilibrium point. Keep up to the second order term. From the equilibrium condition and from the coefficient of the second-order term, you get the equivalent force constant.


ehild

 

[Tanya Sharma]

Have I done it correctly in the OP ? Have I correctly written the EOM for the two charged masses ?Could you help me proceed using the force method .

 

[dauto]

This is just one of a few dozen multi-choice questions on an exam paper? If that's the case, there is probably not a lot of calculation to be done.

You are not told whether in this test the weights are changed from a smaller mass to a larger, or whether it's vice-versa? So even if the period were observed to change, how could you say whether the corresponding ratio is >1 when it might be <1 ?

So, purely on the absence of information allowing you to decide on any contrary option, it seems you are confined to only one answer.


spelling: probably "isolated" spring, and after "changing"

No, they really meant insulated and charging. Look at the picture. The question is whether charging the masses with electric charges q affects the osculation period.

 

[ehild]

Have I done it correctly in the OP ? Have I correctly written the EOM for the two charged masses ?Could you help me proceed using the force method .

It looks correct. Expand the force about the equilibrium point.

ehild

 

[NascentOxygen]

No, they really meant insulated and charging. Look at the picture. The question is whether charging the masses with electric charges q affects the osculation period.

Big oops! I guess I should click on the thumbnail.

 

[Tanya Sharma]

It looks correct. Expand the force about the equilibrium point.

I am not sure what do we mean by equilibrium point in this two body problem ?

At equilibrium k_eq²/(x_2e-x_1e)² = k(x_2e-x_1e-L) ,where x_2eand x_1e are respective equilibrium positions .

Or, k_eq²/x_e² = k(x_e-L) ,where x_e is the separation between the masses at equilibrium.

Is it correct ?EOM for mass m1 is m1d²x1/dt² = kx - k_eq² / (x+L)²

EOM for mass m2 is m2d²x2/dt² = -kx + k_eq² / (x+L)² .

Do I need to expand the 2nd term on the RHS about x_e ?

 

[ehild]

I am not sure what do we mean by equilibrium point in this two body problem ?

At equilibrium k_eq²/(x_2e-x_1e)² = k(x_2e-x_1e-L) ,where x_2eand x_1e are respective equilibrium positions .

Or, k_eq²/x_e² = k(x_e-L) ,where x_e is the separation between the masses at equilibrium.

Is it correct ?


EOM for mass m1 is m1d²x1/dt² = kx - k_eq² / (x+L)²

EOM for mass m2 is m2d²x2/dt² = -kx + k_eq² / (x+L)² .

Do I need to expand the 2nd term on the RHS about x_e ?

Yes, and write the differential equation for x-xe.

ehild

 

[vela]

Do you know about reduced mass and converting a two-body problem into a one-body problem?

 

[Tanya Sharma]

Hello Vela

Do you know about reduced mass and converting a two-body problem into a one-body problem?

Please check my understanding about reduced mass .

In the two body problem ,just as in case 1 where uncharged masses are present and the two masses are under the action of internal spring force ,we may consider mass 1 to be stationary (or treat it as origin) and replace mass 2 with reduced mass μ = m1m2/(m1+m2) .The relative separation 'x' works as the x coordinate of the 2nd mass .

Is this correct ?

I initially intended using the concept of reduced mass ,but there may be something wrong with my implementation . Please have a look at the OP where I have found the time using reduced mass .

What is the correct way of converting this two body problem in one body problem ?

 

[vela]

Yes, that's correct, and the differential equation you derived reflects just what you said. The point is, you didn't need to rederive this result. You could have gone straight to
$$\mu \frac{d^2 x}{dt^2} = -kx.$$ For the charged masses, tack on the electric force and expand to first order to find the effective spring constant.

 

[Tanya Sharma]

You could have gone straight to
$$\mu \frac{d^2 x}{dt^2} = -kx.$$ For the charged masses, tack on the electric force and expand to first order to find the effective spring constant.

$$\mu \frac{d^2 x}{dt^2} = -kx + k_e\frac{q^2}{(x+L)^2}$$

Is this correct ?

 

[vela]

Looks good.

 

[Tanya Sharma]

When x<<L , $$\mu \frac{d^2 x}{dt^2} = -kx + k_e\frac{q^2}{L^2}(1-\frac{2x}{L})$$

$$\mu \frac{d^2 x}{dt^2} +(k+\frac{2k_eq^2}{L^3})x = k_e\frac{q^2}{L^2}$$

Here effective spring constant $K = k+\frac{2k_eq^2}{L^3}$ and time period is $T = 2\pi \sqrt{\frac{\mu}{K}}$

Is this correct ?

Edit :Fixed errors

 

[vela]

Right idea, but you made some algebra mistakes.

 

[Tanya Sharma]

OK..I have fixed the errors .Does post#15 make sense ?

 

[vela]

Yup.

 

[Tanya Sharma]

OK...

Can I conclude that the time period of oscillation of charged masses is less than that of the uncharged masses i.e the correct option is b) >1 ?

 

[ehild]

Yes.

ehild

 

[Tanya Sharma]

Thanks...

Is the time period of md²x/dt² + kx = 0 and md²x/dt² + kx = C same ? Is the time period in both the cases given by 2π√(m/k) ?

 

[TSny]

Is the time period of md²x/dt² + kx = 0 and md²x/dt² + kx = C same ? Is the time period in both the cases given by 2π√(m/k) ?

Yes.

---------------------------

Maybe I'm just being a nuisance, but in your equation $$\mu \frac{d^2 x}{dt^2} = -kx + k_e\frac{q^2}{(x+L)^2}$$ it is not necessarily true that x<< L for small oscillations.

If the spring constant is weak while the charge q is strong, then the electric force can shift the equilibrium position of the system quite a bit. So, in the new equilibrium position, the amount of stretch, x, of the spring from its natural length, L, can be large. It seems to me that you should expand about the new equilibrium position rather than about L. This will modify your answer for the effective spring constant. You can then see if it affects your final conclusion about the period.

 

[ehild]

What is the correct way of converting this two body problem in one body problem ?

Here is the derivation how a two-body problem can be reduced to one body-problem. I write it in one dimension,but the derivation is the same for 3D.

There are two point masses, m₁, m₂. Their coordinates are x₁ and x₂. m₁ exerts force f₁₂ on m₂ and m₂ exerts force f₂₁ on m₁. Assume the forces act along the line connecting the masses.

f₁₂=-f₂₁=F

m_1\ddot x_1=-F.....(1)
m_2\ddot x_2=F.....(2)

Add the equations:
m_1\ddot x_1+m_2\ddot x_2=0....(3)

The coordinate of the centre of mass is X
X=\frac{m_1 x_1+m_2 x_2}{m_1 +m_2}
so you can write equation (3) as (m_1+m_2)\ddot X=0

The centre of mass moves with constant velocity. You can choose the frame of reference fixed to the CM: Then X=0,

m_1x_1+m_2x_2=0.

Divide equation (1) with m₁, equation(2) with m₂ and subtract them.

\ddot x_1=-\frac{F}{m_1}
\ddot x_2=\frac{F}{m_2}
\ddot x_2-\ddot x_1=\frac{F}{m_2}+\frac{F}{m_1}

Denote x₂ -x₁ = x
and
\frac{1}{μ}=\frac{1}{m_1}+\frac{1}{m_2}

You get the equation for the relative coordinate x=x₂-x₁ in terms of the reduced mass μ and the force of interaction F

μ \ddot x= F, the same, as in case of a single body of mass μ.

ehild

 

[Tanya Sharma]

Maybe I'm just being a nuisance

I beg to differ

but in your equation $$\mu \frac{d^2 x}{dt^2} = -kx + k_e\frac{q^2}{(x+L)^2}$$ it is not necessarily true that x<< L for small oscillations.

Do you mean x<< x_e for small oscillations ?

If the spring constant is weak while the charge q is strong, then the electric force can shift the equilibrium position of the system quite a bit. So, in the new equilibrium position, the amount of stretch, x, of the spring from its natural length, L, can be large. It seems to me that you should expand about the new equilibrium position rather than about L. This will modify your answer for the effective spring constant. You can then see if it affects your final conclusion about the period.

Please have a look at post#8 .Should I continue with that ?

 

[ehild]

When x<<L , $$\mu \frac{d^2 x}{dt^2} = -kx + k_e\frac{q^2}{L^2}(1-\frac{2x}{L})$$

$$\mu \frac{d^2 x}{dt^2} +(k+\frac{2k_eq^2}{L^3})x = k_e\frac{q^2}{L^2}$$

Here effective spring constant $K = k+\frac{2k_eq^2}{L^3}$ and time period is $T = 2\pi \sqrt{\frac{\mu}{K}}$

Is this correct ?

You should expand around the equilibrium.

ehild

 

[Tanya Sharma]

Yes, and write the differential equation for x-xe.

ehild

You should have expand around the equilibrium.

ehild

We have two equations ,

k_eq²/x_e² = k(x_e-L)

$$\mu \frac{d^2 x}{dt^2} = -kx + k_e\frac{q^2}{(x+L)^2}$$

Here x_e is equilibrium separation and x is the extension from the equilibrium length ,but what is the relationship between the two ?

Should I substitute the value of L from first equation into the second .

 

[ehild]

Do the Taylor-series expansion around xe. You can assume that x-xe is small with respect to xe+L.
Also note that $$\ddot x = d^2(x-x_e)/dt^2$$

ehild

 

[Tanya Sharma]

Do the Taylor-series expansion around xe. You can assume that x-xe is small with respect to xe+L.
Also note that $$\ddot x = d^2(x-x_e)/dt^2$$

ehild

Sorry...I didn't understand this.I am not good with approximations and taylor expansions .Could you please elaborate on this ?

 

[ehild]

Do you know what Taylor series is? f(x) = f(xe)+f'(xe)(x-xe)+0.5 f''(xe) (x-xe)²...Stop at the first-order term.

xe is constant. So the derivatives of x-xe are the same as the derivatives of x. You can write the differential equation in terms of Δx=x-xe.

ehild

 

[Tanya Sharma]

Do you know what Taylor series is? f(x) = f(xe)+f'(xe)(x-xe)+0.5 f''(xe) (x-xe)²...Stop at the first-order term.

OK...I know the definition but haven't applied it in the problems.

What is f(x) which we need to expand ?