Past exam of the mathematical methods II course of the University of Oxford / 2016 resit / 11 / a

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- Solution a

Solution

The three nearly horizontal segments have lengths

2 a, a, a

. To first order in small transverse slopes, the upward force on the first point mass is

T [(y_{2} - y_{1}) / a - y_{1} / (2 a)]

. The force on the second is

T [(y_{1} - y_{2}) / a - y_{2} / a]

. Applying Newton's second law to masses

m, 2 m

therefore gives

\overset{y}{¨}_{1} + \frac{T}{2 ma} (3 y_{1} - 2 y_{2}) = 0, \overset{y}{¨}_{2} + \frac{T}{2 ma} (2 y_{2} - y_{1}) = 0.

The approximation treats the string as massless, retains the imposed equilibrium tension and neglects gravity, as stipulated.

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