Q. 90

Question

Create a situation different from any found in the text that can be represented by a matrix

Step-by-Step Solution

Verified

Answer

We can determine the kinetic energy of the system using matrices.

1Step 1. Create a situation

Let us consider four point masses $m_{1}, m_{2}, m_{3}, m_{4}$ having velocities $v_{1}, v_{2}, v_{3}$ and $v_{4}$ respectively.

As we know the kinetic energy of any system is given by the formula $K E = \frac{1}{2} m v^{2}$ .

Therefore, to find the kinetic energies of the four masses, we have to convert them into matrix form.

Let the matrix $M$ represents the masses and matrix $V$ represents the velocities.

$\begin{aligned} M = [\begin{array}{cccc} m_{1} & 0 & 0 & 0 \\ 0 & m_{2} & 0 & 0 \\ 0 & 0 & m_{3} & 0 \\ 0 & 0 & 0 & m_{4} \end{array}] \end{aligned} V = [\begin{array}{llll} v_{1} & v_{2} & v_{3} & v_{4} \end{array}]$

2Step 2. Find the Kinetic energy

Therefore,

$K . E = \frac{1}{2} [V] [M] [V]^{T}$

$K . E = \frac{1}{2} [\begin{array}{llll} v_{1} v_{2} v_{3} v_{4} \end{array}] [\begin{array}{cccc} m_{1} & 0 & 0 & 0 \\ 0 & m_{2} & 0 & 0 \\ 0 & 0 & m_{3} & 0 \\ 0 & 0 & 0 & m_{4} \end{array}] [\begin{matrix} v_{1} \\ v_{2} \\ v_{3} \\ v_{4} \end{matrix}] K . E = = \frac{1}{2} [m_{1} v_{1}^{2} + m_{2} v_{2}^{2} + m_{3} v_{3}^{2} + m_{4} v_{4}^{2}]$

Hence, we have determined the kinetic energy of the system using matrices.

Q. 89

Q. 91

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