The force required to compress an “imperfect” horizontal sprin...

Question

The force required to compress an “imperfect” horizontal spring (doesn’t follow Hooke’s law) an amount x is given by F = 150x + 12x³ , where x is in meters and F in newtons. If the spring is compressed 2.0 m, what speed will it give to a 3.0-kg ball held against it and then released?

Accepted Answer

Welcome back. Everyone. In this problem, a tennis ball launcher uses a spring mechanism to project balls, the force to compress the spring and amount to Z is given by F equals 100 Z plus five Z cubed where Z is in meters and F in new times. If the spring is compressed to 0.8 m, what speed will it give to a 0.057 kg tennis ball? When released A says that it will give 25 m per second. B says 29 m per second, C 34 m per second and D 40 m per second. Now, in this problem, if we are trying to figure out the speed that will be given to a tennis ball where we know its force, we know how much the spring has been compressed and we know the mass of the ball. Then we can start to think about the work energy principle. Recall that the work energy principle states that the work done on an object is equal to its change in kinetic energy. In other words, W OK is equal to the change in kinetic energy which we know is equal to half MV squared. Now, we can use this principle to connect the stored energy in the spring to the kinetic energy of the tennis ball upon release to figure out the speed that uh the, the, the spring, the compressed spring will give to the tennis ball. Now, since we found here that our change in kinetic energy is work and that's half MV squared, we can figure out the value of our speed from the work. OK? Because not, no means OK, that since work equals half MV squared, OK. And if we multiply both sides by two and divide by M, that means V squared is going to be equal to two W divided by M, which then means that V if we square root both sides is going to be equal to the square root of two W divided by M. No, we know the mass of our tennis ball here. But the problem is we don't know the work that was done. OK. So we don't need to figure out the work that was done. Let me share that in red here. I put it in blue. And if we can figure out that work, then we will be able to figure out the speed of the bar. Now we're given the force to compress the spin. What do we know about work and force? Well, we know that work is actually the integral of force with respect to the distance. OK. So in this case, since work is the integral of force with respect to distance. In this case, our distance is given by the amount Z that means we can integrate F with respect to Z between the bounds of the spring where it tells us the spring is compressed to 0.8 m. So that's between zero and 0.8 m. And once we find that force with respect to Z, then we should be able to find the velocity. So let's go ahead and do that. So since this is the case, since work is the integral of force, that means we can integrate between zero and 0.8 m or expression for force 100 Z plus five Z cubed. OK. With respect to Z. Now, when we integrate 100 Z, let me add that here. That's going to be 100 Z squared divided by two. OK. Which, well, we can rewrite it later. And when we integrate five Z cubed, that's going to be five Z to the fourth divided by four. And all those are between the boundaries, 0.8 m and zero. Now, if that's the case, that means then that 100 divided by two is 50. So there's really 50 Z squared plus five fourths of Z to the fourth between 0.8 m and zero. Now, here's the thing if we're talking about it from zero, if we substitute zero into our expression, OK. It's really going to be 50 multiplied by zero squared and five multiplied by zero to the fourth divided by four, which we know is going to be is going to be zero. So really then the work to find the work, we can substitute 0.8 m for Z. So that's 50 multiplied by 0.8 m all squared plus five fourths of. So let me write that properly here. Five fourths of 0.8 m raised to the fourth and all of that is going to be subtracting zero, which we know is just going to give us our answer here. Now, when we solve for our work, we can put this expression into our calculator and we're gonna get 32 plus 0.512 which is 32.512 jules. Thus, the work done was 32.512 jules. Now that we know how much work is done, we can substitute it into our formula to solve for the speed. OK? So solving for our speed V, no, that means V is going to be equal to let me get rid of this. Here. Z is going to be equal to the square root of two multiplied by the work done. That's 32.512 Joules put that in bracket here and all of that is being divided by the mass of the ball which is 0.057 kg. So let's go ahead and solve for the speed. Now, when we put this expression into our calculator, OK, then we should find that our speed V is 33.77 m per second. Notice that all of our answers are rounded to the nearest meter per second. So if we do that, we get our velocity or our speed to be 34 m per second. Thus the spring will give the ball a speed of 34 m per second when it is released. C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Struggling with Physics?

The force required to compress an “imperfect” horizontal spring (doesn’t follow Hooke’s law) an amount x is given by F = 150x + 12x³ , where x is in meters and F in newtons. If the spring is compressed 2.0 m, what speed will it give to a 3.0-kg ball held against it and then released?

Watch next

Work By Springs practice set