 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2016/9/a/solution

Each spring force is its extension multiplied by the spring constant, acting to restore the equilibrium. The equations of motion are

m \overset{x}{¨}_{1} = k (x_{2} - x_{1}), M \overset{x}{¨}_{2} = k (x_{1} - 2 x_{2} + x_{3}), m \overset{x}{¨}_{3} = k (x_{2} - x_{3}) .

No end is attached to a wall. The spring extensions are

x_{2} - x_{1}

and

x_{3} - x_{2}

k [(x_{2} - x_{1})^{2} + (x_{3} - x_{2})^{2}] /2

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