Vertical Simple Harmonic Motion

[Tanya Sharma]

Homework Statement

Homework Equations

The Attempt at a Solution

Suppose mass of platform is ##M_1## and that of coin is ##M_2## . The normal force between them is N .

EOM for platform (mass M1) = ##-kx+M_{1}g+N = M_{1}\ddot{x}##

EOM for coin(mass M2) = ##M_{2}g-N = M_{1}\ddot{x}##

This gives , ## N = \frac{kxM_2}{M_1+M_2} ## . The normal force vanishes when x= 0 ,i.e when the spring acquires its natural length .

I feel there is something wrong with my work . I would be grateful if somebody could help me with the problem.

 

[Suraj M]

More than one option right?
You start,! you think Option A is right? or B?
Also i think there might be a simpler way to solve you're problem apart from your posted method, just use ##a = \omega^2 x## and a=g if you get the first part (option A or B ) you'll know what is x.

 

[Chestermiller]

Hi Tanya. You really don't have to include the mass of the platform in the analysis. It is forcing the coin to move in the manner it tells the coin to move. So, the motion of the coin is just

y = Asinωt

From this, what is the acceleration of the coin? What does the force balance on the coin give you? Look at the force balance at 4 times through the period.

Chet

 

[Tanya Sharma]

Hi Chet

Thanks for replying . I think I misunderstood the setup .

Do you agree that both (1) and (3) are correct options ?

 

[Tanya Sharma]

Let us change the setup a bit .

Instead of a platform undergoing SHM at constant angular frequency ##\omega ## , we have a block M1 attached to a vertical spring and block M2 placed on it . Both the blocks are pushed down a small distance 'd' and released . Now if we have to find the position and time where the upper block M2 leaves M1 ,then do you think my analysis is correct in the OP?

Does the upper block leave the lower block when the spring returns to its unstretched length ?

 

[Chestermiller]

Hi Chet

Thanks for replying . I think I misunderstood the setup .

Do you agree that both (1) and (3) are correct options ?

Yes.

 

[Tanya Sharma]

Okay .

What is your opinion on post#5 ?

 

[Chestermiller]

Okay .

What is your opinion on post#5 ?

I think it's not correct. I get the same result as with the original problem statement, except with ω^2=k/(M1+M2). For one thing, your force balance on M2 is incorrect.

Chet

 

[Tanya Sharma]

I think it's not correct.

What is the error ?

I get the same result as with the original problem statement, except with ω^2=k/(M1+M2).

Yes , but till the point they move together . Isn't it ? How would we find the position when M2 leaves M1?

For one thing, your force balance on M2 is incorrect.

It is typo in EOM of M2 . It should be M2 on right side .

 

[Chestermiller]

I don't know where you get your algebraic equation for N.

If I add your two equations together, I get:

$$-kx+(M_1+M_2)g=(M_1+M_2)\frac{d^2x}{dt^2}$$
From this we see that the equilibrium position of the spring is at:

$$x_0=\frac{(M_1+M_2)g}{k}$$

If we define y as the distance relative to the equilibrium position, then ##x=y+x_0##. If we substitute these into the differential equation, we get:

$$-ky=(M_1+M_2)\frac{d^2y}{dt^2}$$

The solution to this equation subject to your prescribed initial conditions is:
$$y=Acosωt$$
where ##ω^2=\frac{k}{(M_1+M_2)}##

The acceleration is then ##y''=-Aω^2cosωt##
If we substitute this into the equation for M2, we obtain:
$$N=M_2(g+Aω^2cosωt)$$
The right hand side of this equation has a minimum value when cosωt=-1. That's when the two masses are at their highest point.

Chet

 

[Tanya Sharma]

Thanks Chet for the effort you have put into help me understand the problem .

But , instead of eliminating N as in the way you have done , if I eliminate ##\ddot{x}## from the two EOM's ,don't we get the normal N as a function of x i.e N(x) = ##
N = \frac{kxM_2}{M_1+M_2}
## ?

From this we get x = 0 , when N=0 .

Please give your views .

 

[SammyS]

The Attempt at a Solution

Suppose mass of platform is ##M_1## and that of coin is ##M_2## . The normal force between them is N .

EOM for platform (mass M1) = ##-kx+M_{1}g+N = M_{1}\ddot{x}##

EOM for coin(mass M2) = ##M_{2}g-N = M_{1}\ddot{x}##

This gives , ## N = \frac{kxM_2}{M_1+M_2} ## . The normal force vanishes when x= 0 ,i.e when the spring acquires its natural length .

I feel there is something wrong with my work . I would be grateful if somebody could help me with the problem.

I don't understand your sign conventions.

Ignoring that, your "EOM" for mass, M₂ in apparently incorrect.

The right hand side should be, ##\ M_2\ddot{x}\ .## That is the result of the net force on the coin.

 

[Chestermiller]

x = 0 is not the equilibrium position. x = 0 is the unstretched length of the spring. It turns out that this is also equal to the highest position of the masses corresponding to the lowest value of A at which separation occurs.

Chet

 

[Tanya Sharma]

x = 0 is not the equilibrium position. x = 0 is the unstretched length of the spring.

When did I say that x= 0 is the equilibrium position ? But then what I found in OP that the blocks separate when the spring regains its natural length is correct .

Right ?

It turns out that this is also equal to the highest position of the masses corresponding to the lowest value of A at which separation occurs.

Chet

How did you get that ?

From the equiibrium position , we push the blocks by a distance 'd' downwards ? Isn't 'd' the amplitude of the SHM ?

 

[Chestermiller]

From my previous analysis,

x = x₀ + y

with ##x_0=\frac{(M_1+M_2)g}{k}##

So,
$$x=\frac{(M_1+M_2)g}{k}+y$$

I also showed that, at the minimum amplitude for separation,

$$y=-A=-\frac{g}{ω^2}=-\frac{(M_1+M_2)g}{k}$$

So,
$$x=\frac{(M_1+M_2)g}{k}-\frac{(M_1+M_2)g}{k}=0$$

So basically, what this says is that you have to pull the masses down to at least 2X the equilibrium displacement in order for separation to occur. And you showed that if you pull the masses down at least that far, the separation will always take place when the spring attains its unextended length.

Chet

 

[ehild]

Homework Statement

Homework Equations

The Attempt at a Solution

Suppose mass of platform is ##M_1## and that of coin is ##M_2## . The normal force between them is N .

EOM for platform (mass M1) = ##-kx+M_{1}g+N = M_{1}\ddot{x}##

The platform is made to perform SHM with angular frequency ω and with increasing amplitude. That means its position is Asin(ωt), a known function of time, with A slowly increasing with time.

EOM for coin(mass M2) = ##M_{2}g-N = M_{1}\ddot{x}##

You meant ##M_{2}g-N = M_{2}\ddot{x}## did you not?
That is correct if you choose "down" positive. But the platform can exert only upward normal force, so it must be positive. The coin stays on the platform and does the same motion as the platform. x for the coim is the same as x for the platform. The acceleration of the platform is ##\ddot x = -\omega^2 x##, so
##N =M_{2}g- M_{2}\ddot{x}=M_{2}(g+\omega^2 x)## . At the top, x = -A, negative. The coin loses contact with the platform if the normal force needed to move it together with the platform should be negative. What does it mean for the amplitude?

 

[Tanya Sharma]

So basically, what this says is that you have to pull the masses down to at least 2X the equilibrium displacement in order for separation to occur.

Chet

Sorry I don't understand this statement . What I have understood is that , only if we push the two blocks a minimum distance ##\frac{g}{ω^2}## , then only the upper block loses contact and that too at the upper extreme location ( i.e when the spring attains its natural length ) .

 

[Tanya Sharma]

Hello ehild

The coin loses contact with the platform if the normal force needed to move it together with the platform should be negative. What does it mean for the amplitude?

It means ## A ≥ \frac{g}{ω^2}## . Right ?

 

[ehild]

Yes, And at what position?

 

[Chestermiller]

Sorry I don't understand this statement . What I have understood is that , only if we push the two blocks a minimum distance ##\frac{g}{ω^2}## , then only the upper block loses contact and that too at the upper extreme location ( i.e when the spring attains its natural length ) .

What I'm saying is that, if you start the masses out out at 1X their equilibrium displacement (that is at the equilibrium displacement below the unstretched spring position), you need to lower the masses to 2X the equilibrium displacement (i.e., an additional 1X the equilibrium displacement below the equilibrium displacement) before the mass M2 can begin to pop up after you release the masses. If you lower it to more than this location before you release, the coin will always pop up, and the location that it will do so is always when the spring attains its natural length (as you showed), which is at or below where it attains its upper extreme location.

I hope this makes sense.

Chet

 

[ehild]

I just wonder what problem did you solve instead the one in the OP? The platform can be moved by a motor so as it performs shm. I do not see spring mentioned in the problem text.

 

[Tanya Sharma]

I just wonder what problem did you solve instead the one in the OP? The platform can be moved by a motor so as it performs shm. I do not see spring mentioned in the problem text.

After reading your post , even I am wondering that there is no mention of spring in the OP , whereas I have always been thinking of platform on top of a vertical spring .

I have always been thinking about the setup (problem) in post#5 . Chet and I have been discussing about this problem .

 

[Chestermiller]

I just wonder what problem did you solve instead the one in the OP? The platform can be moved by a motor so as it performs shm. I do not see spring mentioned in the problem text.

In post #5, Tanya refocused on a different problem that she had defined. The remainder of the posts analyze that new problem.

Chet

 

[ehild]

I see. It would be interesting to see what happens after the normal force becomes zero. If the amplitude is exactly g/w², the normal force is zero only for an instant. Both the platform and the coin have zero speed and the same acceleration g. The coin has no time to detach, it stays in contact. As both the coin and the platform start to move downward, the spring pushes the platform upward, its acceleration would be less than g.
At greater amplitudes, the coin pops up when the length is equal the relaxed length, as you said , and after that, the downward acceleration of the platform is greater than that of the coin, so they really separate for a while. But in case the mass of the coin is comparable with that of the platform, the frequency would change and also the equilibrium position. After reaching the relaxed length again, the acceleration of the platform becomes less that that of the coin, and the coin reaches the platform at some time instant, and collides with it. Tanya's model is quite a complicated one! It is interesting to discuss it, but one has to understand the problem text when writing a test or solving a homework.

 

[SammyS]

In post #5, Tanya refocused on a different problem that she had defined. The remainder of the posts analyze that new problem.

Chet

That's interesting. I have been looking in on this thread from time to time, but I either missed post #5 entirely, or at least missed its significance. Chet had things well in hand so I didn't interfere. (I know - that's a bit unusual.)

It is true that Tanya had included a spring in her analysis right from the beginning in the OP which puzzled me, but now that's cleared up.