Calculate force acting on a Rod with constant mass and varying length

[rashboosh]

Hi all,
This is my first post on this forum. Please state out anything that I have done that does not coincide with the forums rules.

Homework Statement

I have to derive the force acting upon a uncooked spaghetti piece which is being pulled upon by a mass attached to its center for an experiment. The purpose of the experiment is to find the young's modulus of the Spaghetti by investigating the deflection of a spaghetti piece held at fixed points at each end. I do not need assistance in deriving the young's modulus of the spaghetti piece however for further clarification, I intend to derive the young's modulus of the spaghetti by plotting a graph of the force acting on a spaghetti vs the deflection of a spaghetti. I will then use the value for the gradient of the graph to find the young's modulus by combining the three formulas of a free-end beam under strain from the centre, the formula for the moment of inertia of a circular cross-sectional beam, and also the formula for the force.
The mass acting upon the spaghetti will remain as a constant value of 0.075 kg whilst the lengths of the spaghetti piece used will be of 0.06, 0.08, 0.10, 0.12, 0.14, 0.16, 0.18 meters.

Homework Equations

Newton's Second Law F=mg

The Attempt at a Solution

Calculating the force acting upon the spaghetti pieces with Newton's second law F=mg will result in a constant value for the Force which is 0.736N. When plotting this constant value in a graph against the deflection of spaghetti, the line of the graph is vertical and henceforth not linear. I am unable to derive a suitable gradient to be used to calculate the young's modulus for the spaghetti. Is it possible to incorporate the length of spaghetti used into Newton's second law to derive an increasing value for F? So as to create a linear graph?

If it has any relevance in solving this matter, the equation I derived to calculate the young's modulus using the gradient is E=(2L^3)/(48m r^4 ∏.

I appreciate your time and effort in assisting me.

Regards,

 

[haruspex]

It seems obvious that you would plot deflection against length in some way. But if you want a straight line and theory tells you the deflection will not be linear with length then you plot some suitable functions of these variables, not the variables themselves.
In the formula you posted, where does the deflection figure?
Btw, I believe a single strand of spaghetti would be uno spaghetto.

 

[Chestermiller]

You appear to have a simply supported beam with a force acting downward at its center. Please write down for us your equation for the deflection as a function of the force, the moment of inertia, the Young's modulus, and the length.

 

[rashboosh]

Just to provide further insight into this matter, this was a practical, not a textbook question.

@Chestermiller
The deflection of the rod was physically measured with a micrometre. My equation for deriving the young's modulus is of E=(2L^3)/(48mr^4 π). This was derived from the formula for a free-end beam under strain from the centre, the formula for the moment of inertia of a circular cross-sectional beam, the formula for the force (f=mg). I have verified my equation with others doing a somewhat identical task.

@haruspex
I have plotted the avg. deflection for each length recorded under a constant mass against the length of the spaghetti rod. The graph results in a linear shape among the plotted data. I will then implement the gradient of the linear line into the equation to derive the young's modulus of the spaghetti E=(2L^3)/(48mr^4 π). The gradient in the equation is "m". I guess that the deflection is merely a component of the gradient of the linear line. I will plot my gradient from the graph into the equation and see if a suitable value for the young's modulus is derived however in the meantime, I would greatly appreciate any thoughts or ideas anyone has to share. The spaghetti is a full length uncooked san remo spaghetti.

I apologize to all if my above explanations do not make sense.

 

[Chestermiller]

For a simply supported beam loaded at the center, the maximum deflection is given by:

$$\delta=\frac{PL^3}{48EI}$$
where P is the load (=mg) and I is the bending moment of inertia:
$$I=\frac{πR^4}{4}$$

Sutstituting, we have

$$\delta=\frac{mgL^3}{12πER^4}$$

Note that the maximum deflection is proportional to the length of the beam to the 3rd power, rather than the first power. If you plot δ as a function of L³, you should get a straight line through the origin with a slope of
$$slope=\frac{mg}{12πER^4}$$

From this, you can calculate the Young's modulus.

 

[rashboosh]

Thank you for your equation however at this point in the practical, it is assumed that I do not know the value of Young's Modulus however it is a component of your end equation $$slope=\frac{mg}{12πER^4}$$ .
I may be wrong however. Would you be able to elaborate more on what you said?

 

[Chestermiller]

Thank you for your equation however at this point in the practical, it is assumed that I do not know the value of Young's Modulus however it is a component of your end equation $$slope=\frac{mg}{12πER^4}$$ .
I may be wrong however. Would you be able to elaborate more on what you said?

Yes. But, from your measurements, you do know the maximum displacement δ as a function of the length of the beam L, and can plot a graph of the maximum displacement as a function of L³. You can measure or calculate the slope of the line on the plot. You can then use the measured slope to calculate the Young's modulus:

$$E=\frac{mg}{12πR^4(slope)}$$

Make sure you get everything in the correct units. The units of the slope are reciprocal length squared.

 

[rashboosh]

Ok, thanks a lot for your help. I hope I am able to return the favor when the need comes. It is pretty late right now but I will get on to your suggestions and post the results when done.

Many Thanks!