Consider the spring-mass system in Figure 2.4.1. Assume that spring k1 has spring constant k1 = 2 N/m, spring k2 has spring constant k2 = 4 N/m, and spring k3 has spring constant k3 = 3 N/m. Find the equilibrium displacements x1, x2 if forces f1 = 10 N and f2 = 5 N are applied to masses m1 and m2 respectively.

Question

Consider the spring-mass system in Figure 2.4.1. Assume that spring [latex]k_{1}[/latex] has spring constant [latex]k_{1} = 2 \ {N}/{m}[/latex], spring [latex]k_{2}[/latex] has spring constant [latex] k_{2} = 4 \ {N}/{m}[/latex], and spring [latex]k_{3}[/latex] has spring constant [latex]k_{3} = 3 \ {N}/{m}[/latex]. Find the equilibrium displacements [latex]x_{1}, x_{2}[/latex] if forces [latex]f_{1} = 10 \ N[/latex] and [latex]f_{2} = 5 \ N[/latex] are applied to masses [latex]m_{1}[/latex] and [latex]m_{2}[/latex] respectively.

Accepted Answer

Our work above shows us that we just need to solve the system of linear equations

[latex](2 + 4)x_{1} - 4x_{2} = 10 \[/latex] [latex] -4x_{1} + (4 + 3)x_{2} = 5[/latex]

Row reducing the corresponding augmented matrix gives

[latex]\left [ \begin{matrix} 6 & -4 \ -4 & 7 \end{matrix} \left | \begin{matrix} 10 \ 5 \end{matrix} \right. \right ] \thicksim \left [ \begin{matrix} 1 & 0 \ 0 & 1 \end{matrix} \left | \begin{matrix} 45/13 \ 35/13 \end{matrix} \right.\right][/latex]

Hence, the equilibrium displacements are [latex]x_{1} = {45}/{13} \ m[/latex] and [latex]x_{2} = {35}/{13} \ m[/latex].

Question 2.4.1: Consider the spring-mass system in Figure 2.4.1. Assume that...

Related Answered Questions