Trouble with PE exam sample problem 2

[bshadel]

Hi folks,

New to the forum, I’m getting ready to take the mechanical PE exam this Friday. I’ve been studying for a better part of the year and hope for some positive results.

I’ve come across a sample problem where I don’t agree with the solution. Perhaps someone can set me straight. By the way, this problem is from “Six-Minute Solutions for Mechanical PE Exam”.

Problem:
Water balloons are launched from a catapult on the rooftop of a building 50 feet off the ground. Balloons must achieve at least 100 yards (horizontal). Catapult spring constant is 0.5 lbf/in and can be extended no more than 5 inches. Weight of each balloon is 4.5 lbf, and each one takes approximately 1 second to launch. Under these conditions, how many springs in parallel will it take to launch 100 yards?

My solution:
I solved for an initial velocity that would send the balloon 100 yards. I then used energy methods to solve for the total spring constant: 1/2kx^2 = 1/2mv^2. Unfortunately my answer was way off.

Book solution:
The acceleration of the balloon was found from the 1 second launch to the terminal velocity in 5 inches. Knowing that, they said: (mass of balloon x acceleration) = (spring constant x 5 inches).

While the units work out, I don’t agree with this simply because the spring force is not constant and varies with the deflection. This is not the only problem like this…there is another that utilizes this type of solution. Am I missing something???

Any assistance is greatly appreciated.

Thanks,
Bryan.

 

[rb1957]

a couple of questions about your analysis zekeman ...
i think you're saying that a single spring could generate the required force, given a big enough lever?
doesn't that mean that any of the answers would work, given different gearing ?
but you started by saying the energy solutions (saying several thousand springs are needed) are right; even tho' that isn't one of the answer choices.
how far does the balloon travel in the 1 sec "launch" ?
(bearing in mind that the ballon travels 300ft in 1.76sec)

 

[boldfish]

I haven't done the calcs but I think there is a gravitational constant in the KE that is being overlooked.

 

[JStephen]

Using a "speed increasing mechanism" (otherwise known as a catapult arm), the energy solution still holds and the number of springs remains the same. This assumes a massless mechanism; otherwise, the number of springs goes up even more.

Normally, a catapult arm doesn't give a linear relation between object distance and spring distance, and you can't solve equations of motion without knowing that relationship.

 

[zekeman]

Rb1957,
The number of springs would be the same as previously calculated since they depend on the energy equation which remains the same for the ideal case of zero mass mechanism; and since I got 260 as the gear ratio, then the launch distance would be
5/12*260 or about 108 feet during the 1 second launch period followed by the 300 feet of horizontal travel.
The implementation of a practical solution would have to account for the energy in the launch mechanism and as JStephen pointed out, increases the number of springs.
Nobody says that a catapult will be used; with proper gearing a mechanical linear output could be achieved.

 

[zekeman]

Quote from JSteven:
"Normally, a catapult arm doesn't give a linear relation between object distance and spring distance, and you can't solve equations of motion without knowing that relationship."

If the mass is at the end of the boom and rotates 90 degrees from the horizontal until launch, the mass moves along the arc and indeed has a linear relationship with the spring motion, if the mechanism attached to the spring is a rack and pinion.

 

[KiwiMace]

Having sat my FE exam today, I found that to even peruse this posting brought about a post-grief stunned silence somewhat like the hours following a funeral.

Ooh! am I allowed to say that? I did sign box 3!

 

[wfn217]

How was the test?

 

[KiwiMace]

i did the general exam and didn't find it too bad. I'm 8-9 years out of eng school and i did feel for the older guys there. I had a pretty intense month studying. Hope its the end of that exam for me.

 

[JStephen]

Zeke, peruse the internet for a while looking at catapults and see if you see any rack-and-pinion models being made.

 

[zekeman]

Jstephen
I never said that they make catapults with a rack and pinion, only a possibility and have no way of knowing nor do I care. My only point is that there is a linear relationship of the motion at the end of the catapult and the gearing that rotates the final actuator and suggested that if the end of the spring moves along the periphery of the input gear like a rack and pinion or a cable attched to an input winch one would get an overall linear relationship. Indeed launch mechanisms for planes on aircraft carriers (on which I worked on in the 50s) had this type of linearity and of course also a mechanical linear output motion.

Where catapults do not exhibit linearity the problem is solvable albeit with moderate difficulty (after writing the dynamic equations) with present computer methods.

 

[Cockroach]

Wow, what a great problem!

I tried it iteratively and got 8.745 springs, I guess 9 springs would be the answer.

You know the springs got to be reasonable, like a few, perhaps 9 is too high.

I took a single spring, deflected it the said amount at the required spring rate and launched the balloon of known mass straight horizontally. Did the balloon pass a point 100 feet horizontal and 50 feet in the air? If yes, then that spring talley is your answer. If no, then add another spring in parallel, determine the equivalent spring constant of the system and recompute. This is the iterative approach I used, by hand.

There are a lot of answers, few really give you an idea on the number of springs required. Too bad, nice problem!

Kenneth J Hueston, PEng
Principal
Sturni-Hueston Engineering Inc
Edmonton, Alberta Canada

 

[GBor]

Ken,

You just made my day. Way up in this thread I said,

Springs in parallel act together k = k1 + k2 + k3 + ..., and k1 =k2=k3..., so

# springs = 57/6 = 10 roughly

Right or wrong, I suspect that is the way they solved it...

I've stayed out of it since then...listening, but perhaps my PE isn't as dusty as I thought

Garland E. Borowski, PE
Borowski Engineering & Analytical Services, Inc.
Lower Alabama SolidWorks Users Group

 

[rb1957]

kenneth,

GBor got pretty much the same answer,
but why does the energy solution give an answer in 1000s ?

Also, GBor got his answer assuming the balloon took 1 sec to accelerate up to speed. i think if this happens then the balloon doesn't travel the required distance in 1.76 sec.

from energy methods, a spring produces a velocity of ...
v = sqrt(k/m)*x = sqrt(6/(4.5/32.17))*(5/12) = 2.73 ft/sec
we need something like 300ft/1.76sec = 170 ft/sec
(it takes 1.76 sec for an object at rest to fall 50ft)
which would be (170/(5/12))^2*(4.5/32.17)/6 = 3890 springs
(like tlee got)

from force methods, a spring applies a force of 6*(5/12) = 2.5 lb, which would accelerate the balloon at 2.5/(4.5/32.17) = 18ft/sec2 ... but how long is this acceleration applied to the balloon ?

oh, there's a flash of the blindingly obvious ...
the energy solution assumes constant velocity, the force solution assumes constant acceleration
to travel 300ft horizontally in the same time as it takes to fall 50ft the acceleration required is 6g, 193 ft/sec2 which is 10 springs.

so is the spring going to accelerate the balloon or impart a delta v ?

 

[JStephen]

To get 6g laterally the whole distance, you need to accelerate the balloon the entire distance, which violates the maximum spring travel.

 

[rb1957]

but the spring force acts on the balloon, creating the acceleration ... what force acts (later) to slow (decelerate) the balloon ?

the spring force is converted into an inertial force, no?

 

[GregLocock]

Wow. There's a few reputations in tatters here. (Possibly mine after this post).

Let's use some common sense.

A 5 inch spring, with a rate of 1/2 lbf/in (which is, incidentally, softer than a typical rubber band), exerts a maximum force of 2.5 lbf. 10 of them will exert a maximum force of 25 lbf, about the weight of a small sack of potatoes. So let's ignore how springs work and pretend that they apply that force for half their stroke, 2.5 inches.

Now, are you guys seriously proposing that the energy in letting a small sack of potatoes drop by 2.5 inches (bearing in mind that is an overestimate) will accelerate a frozen chicken to something over 100 mph?

If you ignore the 1 second piece of weird data, the energy method takes you to 3888 springs. I suppose by ignoring any one other piece of data in the question will allow a range of other answers. The energy method works because it makes no assumptions about 'average' spring forces, 'constant' accelerations, and other such unlikely concepts in the world of dynamics. It does assume an efficiency of 100%, and a frictionless massless catapult arrangement, but I don't see anybody else including those either.

Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[sreid]

No Greg, your reputation won't be in tatters. In my original read I saw (and still do) see the 1 second as one of those canards to throw off the unwary. Conservation of energy still works unless you are building perpetual motion machines.

 

[rb1957]

i think the constant velocity flight makes more sense, i guess 'cause the balloon would only experience acceleration whilst in contact with the spring (that makes a bit more sense, thinking about it for more than a second), and as others have pointed out the spring force decreases as the travel is reduced, but this is all included in the energy calculation.

 

[RARWOOD]

The important thing is that you need 12 springs with a launch angle of 72 degrees to hit the College of Law Building. A little less spring deflection and you can drop the water ballons on the law students heading to class.

A very interesting problem!!! In particular the above discussion illustrates how different assumptions lead to different solutions.

 

[unclesyd]

Verify part of your result.

http://library.advanced.org/10170/project/index.html