Question 2 ur one is a the big world of computational physics, which is pretty n

Question

Question 2 ur one is a the big world of computational physics, which is pretty neglected in the undergraduate curriculum unless you end up taking Physics 115. showed a computer simulation in class that simulates a row of masses connected by springs. It has another constraint that only the vertical components of the spring forces act on the masses (you can imagine that each mass is mounted to a frictionless vertical track and it can slide up and down but not move left or right). The line in the program that calculates the acceleration on each mass looks like this k/m* (2*y[i] a [i] yli 11 y [i-1]) where k is a spring constant. Describe in words first what this equation means, and second what it has to do with the wave equation. The next two lines are vnew [i] v [i] a [i] *dt new [i] y [i] v [i] *dt. Describe what these lines are doing as we

Explanation / Answer

This is the problem concerned with springs of masses m connected to each other, in the Hooke"s law approx. one can assume the interactions between springs to be restricted to neighbouring spring only,

then force on the ith spring due to its neighbouring left spring i-1 and right spring i+1 ,

let coordinate (or displacement) for the springs denoted by: ith spring by y[i] and same for i-1 and i+1 by

y[i-1] and y[i+1] respectively,

so force on ith springs, F = ma

since, F = -k y = -k *{ (y[i+1] - y[i]) - (y[i] -y[i-1]) }, where the interactions due to the neghbouring springs are only considered.

so acceleration a[i] =  -k *{ (y[i+1] - y[i]) - (y[i] -y[i-1]) } / m = k *{ -2*y[i] + y[i+1] + y[i-1] } / m

this is the acceleration of i th spring due to force exerted on it by the left and right springs.

once acceleration is determined, we can find velocity and position of particle also,

from a[i] at time t1 , we can use it to find velocity as v[i] = u[i] + a[i]*dt

or new velocity as vnew[i] = v[i] + a[i]*dt

once velocity determined we can find position

similarly, position of particle as ynew = y[i] + v[i]*dt

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