Falling Dumbbell

sergiokapone

Homework Statement

Determine force vs angle, that acts on the vertical wall from the falling dumbbell. Dumbbell falls without initial velocity. The mass of each bead of dumbbell is m.

Homework Equations

The problem need to be solved using only Newton's equations by writing them for each of the material points.

The Attempt at a Solution

I do not know how I find the acceleration of the points.

Homework Helper
hi sergiokapone!

start by drawing two free-body diagrams, one for each ball

(call the positions of the two balls x and y, and find how x'' and y'' depend on θ and on each other)

show us what you get

Homework Helper
Gold Member
Determine force vs angle, that acts on the vertical wall

What wall? Is there a diagram missing?

Homework Helper
What wall? Is there a diagram missing?

wot? no wall?

sergiokapone
What wall? Is there a diagram missing?

sergiokapone
start by drawing two free-body diagrams, one for each ball

(call the positions of the two balls x and y, and find how x'' and y'' depend on θ and on each other)

show us what you get

x=lcosα
y=lsinα
l - the length of the conjunction
x2+y2=l2

Homework Helper
good morning!
The problem need to be solved using only Newton's equations by writing them for each of the material points.

x=lcosα
y=lsinα
l - the length of the conjunction
x2+y2=l2

(ahh, it's that way round!)

ok, that's the full diagram

now draw the two free-body diagrams, one for each ball, marking the forces clearly on each

sergiokapone
I solved the problem. But in addition to Newton's laws, I had to use the law of conservation of energy.

In the projections on the direction along the dumbbell:
mgsinα-R=mv2/l
R - reaction from the rod.
x=lcosα
y=lsinα
Differentiating, we find velocity
v=l dα/dt

The law of conservation of energy: mgl=ml2dα/dt2/2+mglsinα

Using Newton's third law, force, that acts on the vertical wall F=Rcosα
or, finally
F=mg(3sinα-2)cosα

How can you do without the law of conservation of energy?

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Homework Helper
Gold Member
x=lcosα
y=lsinα
l - the length of the conjunction
x2+y2=l2
So far so good. What about the forces on the falling end?

Homework Helper
Gold Member

wot? no wall?
Love it! Would he be Chad, Kilroy or Foo to you?

sergiokapone
So far so good. What about the forces on the falling end?
See my previous post, please, where I have already written, it is the reaction force from the connecting rod and the force of gravity.

Homework Helper
How can you do without the law of conservation of energy?

divide the reaction at the wall into Rx and Ry, and call the interaction between the two balls T

do ∑F = 0 for the wall ball, and ∑F = ma for the other ball

show us what you get

Love it! Would he be Chad, Kilroy or Foo to you?

he was (i understand) prolific in and after the second world war here in britain, announcing shortages, but i don't think he usually had a name, or was known by one (though apparently technically he was mr chad)

sergiokapone
divide the reaction at the wall into Rx and Ry, and call the interaction between the two balls T

do ∑F = 0 for the wall ball, and ∑F = ma for the other ball

show us what you get

he was (i understand) prolific in and after the second world war here in britain, announcing shortages, but i don't think he usually had a name, or was known by one (though apparently technically he was mr chad)

OK
1st ball:
Rx=Tcosα
Ry=Tsinα

2nd (falling) ball:
Tcosα=max
Tsinα-mg=may

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Homework Helper
Tcosα=max
Tsinα-mg=may

ok, now write ax and ay in terms of α (or write cosα and sinα in terms of x and y), and solve

sergiokapone
ok, now write ax and ay in terms of α (or write cosα and sinα in terms of x and y), and solve

?

cosα = x/l
sinα = y/l

Concerning what to solve?
I do not understand.
Moreover, a(α) depend on second derivative of α, which is unknown.
ax=-..α lsinα-l.α2cosα

(.. - second time deriv, . - first time deriv )

Homework Helper
might be easier to do it the other way …

mx'' = Tcosα = Tx/l which is a lot easier to solve

(write ' and '' for deriviatives)

sergiokapone
Real exponent?

Homework Helper
uhh?

sergiokapone
x=aexp(sqrt T/ml t)

Homework Helper
oh, hold on, T isn't constant, so you'll have to eliminate T first, before trying to integrate …

that should give you the same equation you'd get if you treated it as a single rigid-body and took moments of forces about the corner

sergiokapone
I went back to the problem.
For the center of mass of the dumbell I have:
## 2m \vec a_{cm} = 2m\vec g + \vec R## (1)
##a_{\tau}=gcos\alpha##
##v_{cm}=-\frac{d\alpha}{dt}l##
##l## - halflength of the dumbell.
##\frac{dv_{cm}}{dt}=gcos\alpha##
or
##v_{cm}{dv_{cm}}=-gcos\alpha d\alpha##
Integrating, I find:
##a_{n}=2g(1-sin\alpha)##
Substituting in equation (1) of ##a_{n}##

##2ma_{n}=2mgsin\alpha-R##
Finally
##R=2mg(3sin\alpha-2)##

Is the factor ##2## near ##mg## is correct?

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Homework Helper
Gold Member
I went back to the problem.
For the center of mass of the dumbell I have:
## 2m \vec a_{cm} = 2m\vec g + \vec R## (1)
##a_{\tau}=gcos\alpha##
##v_{cm}=-\frac{d\alpha}{dt}l##
##l## - halflength of the dumbell.
##\frac{dv_{cm}}{dt}=gcos\alpha##
or
##v_{cm}{dv_{cm}}=-gcos\alpha d\alpha##
Integrating, I find:
##a_{n}=2g(1-sin\alpha)##
Substituting in equation (1) of ##a_{n}##

##2ma_{n}=2mgsin\alpha-R##
Finally
##R=2mg(3sin\alpha-2)##

Is the factor ##2## near ##mg## is correct?
I'm not sure what all your unknowns represent, so I can't follow your logic. The lower mass doesn't play any role, so you can ignore it. It has no horizontal acceleration, and since no radius is given for it you have to ignore any KE for it.
I don't get the factor of 2.

sergiokapone
The lower mass doesn't play any role, so you can ignore it.
But what about the law of motion cernter of mass? In this law we need use total mass of the dumbell. Looks like paradox.

Homework Helper
Gold Member
But what about the law of motion cernter of mass? In this law we need use total mass of the dumbell. Looks like paradox.
You don't have to treat the dumbbell as a single object. You can treat each end mass and the bar joining them as separate objects.

sergiokapone
You can treat each end mass and the bar joining them as separate objects.
Then I need to include a unknown force from the joinig bar in equation for the falling mass.
##m\frac{d\vec v}{dt} = m\vec g + \vec F_{bar\,reaction}##

sergiokapone
And I basically want to understand how to apply the equation of motion of the center of mass. Therefore I want
to consider dumbell as a single object .

Homework Helper
Gold Member
And I basically want to understand how to apply the equation of motion of the center of mass. Therefore I want
to consider dumbell as a single object .
Ok, but as I said you need to define all your variables. If ##a_\tau## is the tangential acceleration of the mass centre, how do you calculate that it equals ##g\cos(\alpha)##?

sergiokapone
Ok, but as I said you need to define all your variables. If a_\tau is the tangential acceleration of the mass centre, how do you calculate that it equals ##gcos(α)g\cos\alpha## ?

May be, it was my mistake. I think the correct answer is:
##a_{\tau_{CM}}=g/2 \cos\alpha##
Using the law of motion of the center of mass: ##2m\vec a_{CM}=2m\vec g + \vec R## in projection to the tangential direction (##\vec R## - acts along the bar)
##2ma_{\tau_{CM}}=2mg \cos\alpha##
I get ##a_{\tau_{CM}}=g \cos\alpha##

sergiokapone
##\vec R## - acts along the bar
I think, this is my primary mistake.

Homework Helper
Gold Member
May be, it was my mistake. I think the correct answer is:
##a_{\tau_{CM}}=g/2 \cos\alpha##
Using the law of motion of the center of mass: ##2m\vec a_{CM}=2m\vec g + \vec R##
In that equation, R is the total reaction from the ground. This is not the same as the force along the bar. The difference is the weight of the lower mass.

sergiokapone
This is not the same as the force along the bar.
Yes, I understand.
Correct equation is
##2m\vec a_{ CM}=2m\vec g +\vec N_1 + ⃗\vec N_2 ##
##\vec N_1## - is the reaction from the floore
##\vec N_2## - is the reaction from the wall
Also, I know from the energy law conservation or other the normal and tangential accelerations of the CM. How I can find ##N_2##, I need ##a_x##?