Physics Problems using eulers method

PHYS 411 Numerical Modeling Extra Credit Project Theory Describing the motion of a particle – its position, velocity and acceleration as a function of time – can be achieved fairly easily if the system is not too complex. However, nature quickly can become very complex, so much that an analytical solution (an equation for position or velocity as a function of time) can be too mathematically complex to attain. Numerical modeling techniques can often be used to solve such systems. Most of these numerical modeling methods involve taking a rather complicated system – perhaps one in which the acceleration varies with position, speed or time – and breaking into small intervals (of distance or time) and assuming that the acceleration and /or velocity are constant over each small interval. The smaller the interval, the closer the approximation is to reality. The Euler Method is a numerical modeling method that breaks a complicated motion into discrete intervals of time, and assumes that the acceleration and velocity are constant over each interval. The relation between the acceleration and velocities for some small time interval ?t are as follows: v v a t v t t v t a t t t v t t v t t v a t n n n ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?1 ?1 ( ) ( ) ( ) ( ) ( ) ( ) (This equation is exact if the acceleration is constant.) We can similarly approximate the position after the time interval ?t as well. x x v t x t t x t v t t t x t t x t t x v t n n n ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?1 ?1 ( ) ( ) ( ) ( ) ( ) ( ) The familiar 2 2 1 a(?t) term is missing because of the assumption that ?t is small. If ?t is small, (?t)2 is very small. The position and velocity of the object as a function of time can then be approximated by a series of calculations as follows: Step Time Position Velocity Acceleration 0 to xo vo m F t a o o ? ? ( ) 1 t t t o ? ? ? 1 x x v t o o? ? ? 1 v v a t o o? ? ? 1 m F t a ? ? ( ) 1 1 2 t ? t ? ?t 2 1 x ? x ? v ?t 2 1 1 v ? v ? a ?t 2 1 1 m F t a ? ? ( ) 2 2 . . . n . . . t t t n n ? ? ? ?1 . . . x x v t n n n ? ? ? ?1 ?1 . . . v v a t n n n ? ? ? ?1 ?1 . . . m F t a n n ? ? ( ) The Assignment You may work in groups of up to three students if you desire. Turn in one assignment per group. a) (5 pts) Consider a mass sliding down a frictionless curve in the shape of a quarter circle of radius 2.00 m as in the diagram. Assuming it starts from rest, use Euler’s method to approximate both the time it takes to reach the bottom of the curve and its speed at the bottom. You may either use a spreadsheet like MS Excel or you may write and execute a computer program in the language of your choice. Do three trials: ?t = 0.2s, ?t = 0.02s, and ?t = 0.002 s. Compare the predicted speed at the bottom for each case to the accepted value of 6.261 m/s. (Calculate a percent error.) Does the approximation improve as ?t becomes smaller? b) (5 pts) Repeat (a), but this time assume a constant kinetic friction coefficient of ??k = 0.200. Again determine the time to the bottom and the speed at the bottom. . You need only run one trial: ?t = 0.002 s. As you do not have a “correct” value to compare, do not calculate a percent error. m v = ? 2 m m ? For both (a) and (b), turn in the spreadsheet file or the code and via the D2L dropbox. If you write a program, you’ll need to bring a working computer with the program to my office and demonstrate it. Include in a summary of your results. Hints: (a) Define the position and acceleration of the mass in terms of the angular position ?. Be careful – some computer/spreadsheet functions assume radians. (b) Because it is moving along a circular track, the mass will experience a centripetal acceleration. This will affect the normal force.