Spring force - does it scale linearly?

[denver75]

I'm struggling to get my mind around this concept:

If you have a spring with a known spring constant, and you put a specific mass on it, compress it a certain distance and then release it, can you predict the distance it will travel with the information supplied?

Or in other words, can I calculate where my slingshot will land if I know the spring constant?


Second question:
If I have all the information, can I scale the slingshot up or down, changing the spring and weight to get the same trajectory? And is the spring constant a linear equation?

Thanks!

 

[Nabeshin]

Question 1: Yes. Assuming you know the spring constant, you know the energy stored, and conservation of energy gives you a velocity. From the velocity, basic kinematics create a trajectory.

Question 2: Yes. Ignoring air resistance, the final velocity squared is proportional to the spring constant, the square of the compression, and inversely proportional the the mass. If air resistance is ignored, the final velocity (and angle) completely determine the trajectory, so any of these 3 parameters can be altered to give the same final velocity, and the same equation.

Question 3: I'm not sure why you didn't give more weight to this one, but are you asking if the spring force scales linearly with displacement? (The spring constant itself isn't an equation, this has no meaning) If that was your question, then yes in situations where Hooke's law is valid it does scale linearly. However, any experience with a rubber band will tell you that force doesn't increase linearly (For example, at a certain point the more you stretch a rubber band, the opposing force increases greatly to resist further stretching, until the material can no longer withstand the force and the band snaps). In situations such as these, Hooke's law isn't valid and the situation gets immensely more complicated. But for simple spring-loaded projectiles, it's usually a very good approximation to say the force scales linearly with compression.

 

[denver75]

Thanks for the info, Nabeshin.

Here's a quick follow up question:
If we know that an object that weighs 50 lbs is launched 100' with a big catapult (angle: 41º, velocity: 39mph), is there an easy way to calculate the force generated, and then apply that data to determine the trajectory of another object of different mass using the same catapult and angle?

Thanks!

 

[Nabeshin]

Thanks for the info, Nabeshin.

Here's a quick follow up question:
If we know that an object that weighs 50 lbs is launched 100' with a big catapult (angle: 41º, velocity: 39mph), is there an easy way to calculate the force generated, and then apply that data to determine the trajectory of another object of different mass using the same catapult and angle?

Thanks!

I'm not too keen on the real life uses and workings of catapults, but I'll give my input for a somewhat idealized situation:

Assuming the catapult works independent of the ammo loaded (mass is irrelevant), it will give the same amount of energy to any object launched. You can calculate the kinetic energy of your 50lb rock by 1/2 m v^2. This is the total energy the catapult gives to its ammunition. So, if you launch a 10lb rock you can solve for the velocity of the new mass using the fact that it will have the same amount of kinetic energy. From that you can create a ballistic trajectory.

(Keep in mind this is probably oversimplified. If anyone knows better how real catapults work and can revise this, please do so)

Forces are difficult to deal with because a lot of the time they're not constant and the mechanics involved get complicated fast. So, using energy considerations is often a much better method. Especially if time isn't involved (situations in which you just know initial and final conditions, say).

 

[denver75]

Nabeshin, hugely helpful. That's just the info I needed. Thanks.