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# lab15 - LAB#15 Spring Constant and Simple Harmonic...

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LAB #15 Spring Constant and Simple Harmonic Motion (SHM) Introduction : Within certain limits: a spring obeys Hooke's law when applied to a spring, this law states that the restoring force exerted by the spring is proportional to the change in the length of the spring and in the opposite direction to the displacement of the end of the spring. We express this as: f = -kx (1) where x is understood to be the change in spring length from its unstretched state. We will determine the constant of elasticity k of a spring and determine the period of oscillation (t) of the spring under various loads. We can also find the springs frequency (f). In the case of a coiled spring the equation is only true after sufficient force has been applied to separate all the coils. If a mass of M is hung from the end of a spring which is considered massless and the system is set into vibration, the equation of motion is: F= -kx = Ma (2) k = ! F 2 ! F 1 x 2 ! x 1 ( N M ) (3) and k = ! F ! x Since a = d 2 x dt 2 This equation can be written: d 2 x dt 2 = ! k M x In actual practice we should not overlook the fact that a real spring has mass itself. When m for the spring is included it can be shown that the period becomes this equation for simple harmonic motion, with the period of oscillation given by: T = 2 ! M + m 3 k (4) M is the mass of the object and m is the mass of the spring. The theoretical values for the springs we use are between 9.5 and 10.6 N/m.

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