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Damping and Resonance Problem : Forces and Displacement

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A seismograph is a scientific instrument that is used to detect earthquakes. A simple model of a seismograph is shown below. It consists of a particle of mass m to which a pointer is attached. The particle is suspended by a spring of natural length lo and stiffness k and a damper of damping constant r from a platform of height d which is fixed to the Earth. Let the vertical displacements of the Earth, relative to a fixed origin 0, be denoted by y and let the length of the spring and the damper be x, as shown in the following diagram.

(a) Draw a force diagram showing all the forces acting on the particle.
(b) Express the forces acting on the particle in terms of the given variable and parameters.
(c) Show that the displacement x(t) of the pointer with respect to the platform satisfies the differential equation.

I think I have managed to do parts (a) and (b) successfully (but could do with a confidence check!) but had a problem with part (c):

I cannot figure out where the my: comes from on the RHS of the equation. I am taking the x origin from the top of the device and pointing downwards, which I am unsure is correct since the y origin points upwards. I can reach the equation by, when using Newton's Second Law (F=ma), I take the acceleration to be the sum of x: and y:, thus expanding to give the correct answer. But I have a feeling I am just making this up and will not gain full marks for the question ...

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Forces and displacement are investigated.

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