Problem with spring + inclined plane + conversation of energy etc

[-sandro-]

Homework Statement

A ball (considered as a point) of mass m = 1.5kg is launched from a spring with a Δx= 5cm. Find the elastic constant K knowing that the ball is forced to travel the path ABC and that at point C its speed is zero. AB has no friction while BC has a friction μ_k=0.05.
At point C the ball falls downwards with an air resistance of B = 0.5 N*s/m. What is the weight measured by a scale at point E? What is the distance EF?

This is the drawing
http://imageshack.us/a/img191/1360/55sp.jpg

Relevant equations
E₀ W_nc = E_f
F_f = -βv

The Attempt at a Solution

I'm not sure how to actually solve this problem, I'm not sure about the process.
I assumed that at A there's an elastic potential energy, at B gravitational potential energy + kinematic energy and at C all the energy has been "consumed" by the work done by the force of friction since the speed is 0.

So I tried to set something like this

1/2kx² + W_f = mgh + 1/2mv²

Is this correct? At this point I would solve for K but I don't have the speed for v^{2[/SUP. Plus where is the kinematic energy? Where does it start? Also I don't understand if to find K I must consider the all ABC path or AB is sufficient. Can you also help find out how to solve the other parts with air resistance and that curved path at the end? Even if you don't want to give the answer at least knowing the equations to use and laws behind them.}

 

[barryj]

I haven't worked the problem but here are some things to consider.
The vertical component of the initial velocity at A will be converted into potential energy at B.
The horizontal velocity at A will be converted into frictional work as the mass slides from B to C

However, I wonder if the mass will stay on the incline and then not overshoot the BC section?

 

[-sandro-]

It will but at point C the speed is 0. I think that my professor made it that way to make things easier for the free fall part also for knowing that friction used all the energy.
So I don't have to consider that kinect energy?
Initial Energy: 1/2kx²
Final Energy: mgh (there's no kinetic energy at C)

So the equation would become

1/2kx² -W_nc = mgh

I put a negative sign since the work done by friction is on the opposite side of the direction.
Does this seem correct?

I just noticed I wrote conversation instead on conservation in the thread title :D

 

[barryj]

"1/2kx2 -Wbc = mgh" This seems correct to me.

 

[-sandro-]

I get a value of K = 27048N/m...isn't this too high? maybe wrong calculations

 

[barryj]

That is what I got also.

 

[-sandro-]

OK, sorry but this is the first time I use elastic potential energy. Any help about the following questions? I have no idea how to proceed. And thank you"!

 

[barryj]

If the mass is going horizontally at point E, then the weight should be mg. Now you have to find the velocity at E in order to find out how far the mass will travel to point F. If it were not for the factor, " = 0.5 N-sec/m you could use the change in potential energy from C to E to calculate the velocity. But since you have this drag factor, I would think you would have to do a little integration to find the work done on the air as the mass dropped. Is this something you are familiar with?

 

[-sandro-]

I'm a little familiar with the drag force, so I have to find the velocity at point E? Is weight measured affected at all by the velocity? I find it odd the professor i just asking for a simple "m times g" :D I think the part I can't really solve is the curved path where I'm given the R (i suppose radius?) of the circumference. Will the speed change from D to E?

 

[barryj]

Getting complicated, yes? If the weight is measured on the horizontal part, then the velocity will not matter. However, if the weight is on the curved part, then centripetal force would matter. Another question is whether the drag force exists between D and R or just between C and D?

I expect one of the physics experts to step in here shortly.

 

[-sandro-]

It's only between C e D then the ball will follow the curved path with no external force.
In what way does centripetal force affect the speed of the object when it goes to E?

 

[barryj]

Then you will have to find the velocity of the mass at D taking into account the drag. Now, the question is, is the weight measured just before or after the mass passes the transition point whee it is on the curved surface or the horizontal surface. The difference will be the centripital force that goes away after the mass starts traveling horizontally.

 

[-sandro-]

Reviewing the drawing it seems like the scale is positioned at the end of the curved surface so the centripetal force will matter to measure the weight. Is that what you meant? Basically I'm getting that by weight the question is asking what is the F of the centripetal force. Weight = Force = ma = mv²/r. Did I get it? ;)

The question remains, if I know the initial velocity at D how do I find the final velocity at E to get the Weight (force) and finally the final path EF? something to do with the 90° angle?

 

[barryj]

I think the weight just before the mass is traveling horizontal is the centripetal force, mv^2/r, plus mg then after going horizontal, the weight is just mg.

As to the velocity at D, I am still trying to figure this out. It seems like you might have a differentiable equation to solve unless you already have a formula available.

 

[-sandro-]

humm I think I just got it wrong. I thought that having a centripetal acceleration would increase the tangential velocity from D to E but that actually keeps it constant ( uniform circular motion) so If I have the initial velocity at D (considering the drag force) so the weight at E would the centripetal force + mg, like you said!

 

[barryj]

So to find the velocity at D, you must find an expression for he velocity as a function of time, then do an integration to find distance as a function of time, set this equal to the distance of 2m, find the time, plug in and get the velocity. There may be other ways but this is what I did.

 

[-sandro-]

Thank you! about the curved path, doesn't sliding a circular path downwards increase its Angelita speed due to gravity?

 

[barryj]

Yes it does, but you can use the change in potential energy between point D and E to determine the increase in kinetic energy from D to E and once you have the KE you can find the velocity.

 

[-sandro-]

I'm having problems with finding the velocity at D. At class we've been told what the drag force is and how to find the terminal velocity of a point mass object which g/C_d that would equal to 19.6m/s. But how do I get the speed after 2m knowing the terminal velocity?

 

[barryj]

I am not sure how you learned this topic but what we have here is a case where the drag is proportional to the velocity. This is not always the case and often the drag is proportional to v^2.

In this case, the velocity of a dropped object can be written as v = (mg/B)(1-e^(-tB/m))
I hope this looks famaliar. This shows the velocity of a dropped object starting a 0 and approaching exponentially a terminal velocity of (mg/B).

So, you need a relationship between the velocity and distance as the mass drops 2 meters. What I did was to integrate the velocity to get distance, paying attention to the constant of integration. Now I have distance as a function of time and can fine the time to drop 2 meters. Knowing the time, I can plug back into the velocity equation to get the velocity.

There may be an easier way to get to this point but this will work. If you neglect drag, you can find the velocity at D to be about 6 m/s. With drag it is a little less.

 

[haruspex]

If the mass is going horizontally at point E, then the weight should be mg.

That doesn't quite follow.
The relationship of the scale's position to the curve is not entirely clear. If it is supposed to be immediately after it has straightened out then yes, it's mg. But if immediately before straightening then you need to consider the centripetal force. Note that 'travelling horizontally' doesn't quite fix it: that only specifies the velocity, not the acceleration.
I have another quibble on this problem. The path through B has an abrupt change of direction. In principle, that constitutes an impact, not conserving work. But I would guess you're supposed to assume that at some microscopic scale it is a smooth transition.
Wrt the path CDE, does the air resistance only operate over CD? If it operates over DE as well, you'll need a different differential equation there.

 

[barryj]

Some of these issues have been discussed. See posts 10 thru 13.

 

[haruspex]

Some of these issues have been discussed. See posts 10 thru 13.

Yes, but that discussion kept referring to curved versus horizontal. That isn't the relevant distinction. Even if the curve continued into an upward portion it would be horizontal at the bottom. The question is whether the scale is considered to be part of the curve or part of the straight.
I did miss the response that there's no drag after D.

 

[barryj]

"The question is whether the scale is considered to be part of the curve or part of the straight." The diagram shows the curved part and the horizontal part. Yes, it does make a difference where the scale is placed. I am waiting to see if Sandro got the velocity at D correct.

 

[-sandro-]

I haven't yet because I wasn't able to get the equation for distance in respect to time to find the time and plug it into the velocity equation. Also I'm very confused about final speed at E after DE, I know that at point D the centripetal force will be mV^2/R being V the speed found after the fall with air resistance. Since this object will follow a curved path downwards it will accelerate due to force of gravity (assuming that's the only cause for tangential speed acceleration) but I have no idea how these two forces (centripetal and gravity) interact with each other. I only studied rotational motion on a horizontal plane.On second thought barryj you suggested to find the speed at E with the difference in potential energy? At point D the ball has a kinect energy from the fall of 1/2mvi^2 + some potential energy mgh and at point E it's only 1/2mvf^2 (vi = initial speed, vf= final speed). You mean just solve for Vf? If that's correct it would help me find the weight at point E cause If I have the speed at point E I could substitute that in the centripetal force equation without worrying about the acceleration caused by the force of gravity since it will be included using the change in energy.
Did I get it ? :D

Anyway the scale is definitely included in the curved path otherwise there wouldn't be no reason to ask for the weight at point E if it was just "mg"

 

[barryj]

At point D, you have KE and PE, so at point E, the KE will be increased by the change in PE from D to E. Now that you have the KE at E you can find the velocity at E.

Just before E, the weight will be the sum of the centripital force as determined by the velocity you calculated above + mg. Just after E, the weight will be only mg. Now kowing the velocity, you can probably fined the distance to stop.

Did you get the velocity at D?

 

[-sandro-]

I didn't can you please tell me what equation you used to find the time to fall to point D?

For the change in ME from D to E seems like you're confirming what I already said, good!:D

 

[barryj]

OK, so think about this. If there wassn't any air drag then you can easily find the velocity at D given that the change in PE from C to D is converted into KE. Now with air drag, the velocity at E will be a little slower. If you are into doing differential equations, then you can set up the problem and solve. However there is an already computed formula that works when the drag is proportional to V as opposed to V^2. In this case the Drag is proportional to V.

The equation is.. V =(mg/b)(1-e^(-tb/m)) perhaps you have seen this equation. Notice that as t becomes very large, V becomes the terminal velocity of mg/b.

You don't have t however so if you integrate v(t) you get S(t). So first knowing S = 2 m, you can find t ujsing the integration you just did. Then knowing t you can find v. When you integrate v(t) to get S(t) don't forget to find the constant of integration using the fact that at t = 0, S(t) = 0.