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Resonance of a spring

            
             Hooke's Law states that the force exerted by a simple harmonic oscillator (SHO) is directly proportional to its distortion. In other words:.
             ,.
             where x, is the distortion of the SHO, F is the force it exerts, and k is an elastic constant for the SHO. A spring for instance is a SHO. A spring with a spring constant (k) of 2000 Newtons per meter distorted 0.5 meters, exerts a force of 1000 Newtons.
             Each SHO has what is called a natural frequency. The natural frequency is the frequency at which the SHO oscillates when set into motion. The equation for natural frequency is as follows:.
             ,.
             where m is the mass attached to the spring, and k is the spring constant.
             The purpose of this experiment is to determine the theoretical natural frequency of a given spring, and then determine its natural frequency experimentally, and determine the error between the two. Another purpose is to determine the effect of the frequency of the driving force on the amplitude of the mass's oscillation. The hypothesis is that the experimentally determined natural frequency will be lower than the theoretical natural frequency. Energy is not completely conserved in this lab due to heat and air resistance; therefore the experimentally determined natural frequency will be dampened.
             The first step is to determine the spring constant of the given spring. To do so, a known mass is hung vertically from the spring, and the spring's distortion is measured. From the free-body diagram, it can be seen that the weight of the mass is equivalent to the force exerted by the spring.
             .
             Now that k and m are known, it is possible to calculate the spring's theoretical natural frequency.
             .
             To determine the natural frequency experimentally, the mass is allowed to bounce for five periods while it is timed:.
             .
             Mass (kg) Äx (m) k (N/m) Time (sec) No. Of Osc. fN predicted (1/sec) fN measured (1/sec) Percent Error ùd (1/sec).
             010 0.


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