# System of springs: Energy stored in different springs

You can assume that air resistance and other frictional forces can be ignored in this question

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a) Define a datum for the gravitational potential energy of particle P.

b) Write down the gravitational potential energy of particle P at a general point of motion.

c) Write down the kinetic energy of a particle Pat the same instant

d) Determine the potential energy stored n each spring at the same instant.

e) Write down an expression representing the total mechanical energy of the system at the same instant.

https://brainmass.com/physics/energy/system-springs-energy-stored-different-springs-578075

## SOLUTION This solution is **FREE** courtesy of BrainMass!

Please see attached file.

(a)

For all such problems the gravitational potential energy means the difference in potential energy of the particle from that at a particular reference level. Mostly the reference level is taken as the surface of earth and the gravitational potential energy of a particle of mass m, at a height h above the surface of earth, is given by m*g*h.

Here we may take the top level along A as reference level as i (unit vector) is shown downward and the distance x is measured from that level (thus reference level is x = 0)

(b)

Here the position x is the general point (position) during the motion and as this point is x below our reference level the gravitational potential energy of the particle at this position is given by

(c) If the velocity of the particle at this position is v than its kinetic energy is given by

(d)

The potential energy stored in a spring is given by

Here k is the spring constant and l is the extension (or compression) in the spring.

The length of the first spring of force constant 2k in this position is x hence the extension in the spring is

( is the unscratched length of the springs)

Thus the elastic potential energy stored in the first spring of constant 2k at this instant will be

Similarly the length of spring of spring constant 3k is and thus its extension will be

Thus its energy

And the length of spring of spring constant 4k is and thus its extension will be

Thus its energy

(e) The total energy of the system at this instant is the sum of gravitational, kinetic and potential energies and hence given by

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