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where x is the instantaneous velocity of the mass.

The force in the spring is kx; work done by the spring is kx Sx. The potential energy is the work done by this force and is given by

Jo 2

The total energy in the system is a constant. Thus

Differentiating with respect to x, we get mxx + kxx = 0 (2.59)

Since x cannot be zero for all values of t, we get mx + kx = 0 (2.60)

which has the same form as Eq. (2.52). This differential equation has a solution of the form x = A sin(wt + a) (2.61)

where A is the maximum displacement and wA is the maximum velocity. Maximum kinetic energy is given by

Maximum potential energy is

or which is the natural frequency of the simple oscillator. This method, in which the natural frequency is obtained by equating maximum kinetic energy and maximum potential energy, is known as Rayleigh's method.

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Renewable Energy 101

Renewable Energy 101

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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