# Spring constant of two cords

#### colonel

Imagine you have two chords attached to a softball holder. The spring constant is 300, measured when the chords were 10 cm apart. Now if the distance between the unattached ends were increased, does the spring constant change?
Also, say the distance between the chords are increased, the holder is pulled back and down (3-D)how will I measure the "x" value for the equation E= 1/2 kx^2?

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#### arcnets

Hi colonel,
I'm afraid I don't quite understand the problem. Are the chords meant to be vertical in the original position? Is 10 cm the diameter of the softball holder?

#### colonel

There is a plastic funnel for the softball. Attached to the funnel are two surgical chords, about 1 m in length each. The funnel and the two chords form the "slingshot". To measure the spring constant, I hung the "slingshot" from a bracket on the wall. In this position, the two chords were 10 cm apart (same width as bracket). To launch the softball, there are two people holding onto the chords, one person per chord. But you cannot have the two people standing 10 cm apart as they will get whacked when the softball is released. The two people spread apart a little, say both go 1 m. The third person pulls the funnel (with the softball) back and down, and releases it. As you can see, there are 3 dimensions here : each chord goes 1 m away from an imaginary centre, back 1m from the pull, and down 1 m from the pull. Does this change the spring constant and how would I determine the "x" value for the equation E = 1/2 kx^2? Once again, thank-you for your time.

#### gnome

Colonel, you have a big problem. The spring constant in Hooke's Law lets you calculate the force exerted by a spring in the direction opposite to the displacement (stretching) of the spring -- opposite in direction but along the same line as the spring. That is what you measured by hanging the "slingshot" on the wall, presumably with some known weight in the cup.

But that is very different from the forces involved in your final "apparatus". There, each piece of cord [not chord :)] exerts a force on the cup based on the spring constant of the tubing, along the line of the tubing in a direction back towards the person holding the end of the cord. But the resultant (net) force on the ball is in the direction midway between the two "anchor" people. So, if &theta; is the angle between the line of one side of the cord and the line of flight of the ball, and x is the distance that each side of the cord is stretched from its natural length, the total force exerted on the ball by the cord at the instant that the third person releases it is 2*k*x*cos&theta;. That's the force you are trying to determine, and it has to have a different spring constant than the one you measured. And frankly, I'm not sure there actually exists a spring "constant" for your setup. The concept of the spring constant is that the force exerted by the spring is linearly proportional to the displacement of the spring. Here, the variation in the amount of net force in the direction of flight is related to the stretching of the cord and to the angle (which is also constantly changing) between the cord and the line of flight. I'm not sure, but I doubt if the resulting relationship is linear.

There's another problem. Your "spring" is not just the rubber tubing. Your "spring" includes the two people holding the ends of the cord, and the tension in their muscles will definitely play a very significant role in the propulsion of the ball. When the third person releases it, those two people will suddenly jerk in the direction of flight; it's almost inconceivable that they will be able to avoid doing so. Therefore, their muscles will also be contributing to the force that launches the ball. Your experiment would be more accurate if you could replace them by two poles anchored securely in the ground & simply tie the cord to the poles.

Either way, you should re-measure your spring constant with your apparatus configured exactly the way it will be set up for the actual trial. For example, have the "puller" use a spring balance to pull the funnel back several measured distances & record the amount of force needed in each case. If you do this for several different stretch distances you may be able to determine if there exists a linear (or approximately linear) relationship.

By the way, if all you are really trying to do is determine the potential energy of the system, there is a much easier, and probably much more accurate way, that doesn't involve that formula U = (1/2)kx2. Think about it.

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