The four blocks of different masses mi are connected by ropes of negligible mass

Question

The four blocks of different masses mi are connected by ropes of negligible mass. Three of the blocks lie on a inclined plane, the coefficients of friction between the blocks and the plane being ui. The equations of motion for the blocks can be shwon to be.

t1 + m1a = m1g(sin(theta) - u1cost(theta))

-t1 + t2 + m2a = m2g(sin(theta) - u2cos(theta))

-t2 + t3 + m3a = m3g(sin(theta) - u3cos(theta))

-t3 + m4a = -m4g

Where ti denotes the tensile forces in the ropes and a is the acceleration of the system. Determine a and ti if theta = 45 degrees, g = 9.82 m/s^2 and

m = [10   4    5    6]^T kg

u = [0.25     0.3    0.2]^T

Do this in Python 3.3

Explanation / Answer

import numpy as np

import math

m=[10,4,5,6]

u=[0.25,0.3,0.2]

k=45

a = np.array([[1,0,0,m[0]], [-1,1,0,m[1]], [0,-1,1,m[2]], [0,0,-1,m[3]]])

b = np.array([9.8*m[0]*(math.sin(k)-u[0]*math.cos(k)), 9.8*m[1]*(math.sin(k)-u[1]*math.cos(k)), 9.8*m[2]*(math.sin(k)-u[2]*math.cos(k)), -m[3]*9.8])

x = np.linalg.solve(a, b)

print(x)

Answer= [ 40.34139449 55.44832119 76.90605731 3.01767622]

you need to install numpty to solve this

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