Question

Asked Nov 25, 2019

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*We will now determine how the 1/3 rule comes about. *

*Consider a spring of mass m _{s} which is attached to a wall and oscillates on a frictionless surface as shown below. The spring’s mass is uniformly distributed along the length of the spring. *

*We will start with the infinitesimal form of kinetic energy, i.e. dKE = ½ (dm _{s })v^{2}. This formula will apply to an infinitesimal segment of the spring of length dx and mass dm_{s }as indicated below. *

*For any point on the spring, the velocity of oscillation will be given by v = (v _{e}/L)x where v_{e} is the velocity of the spring at its end where the mass m is attached, and L is the stretched length of the spring at that instant. Thus, when x = 0 then v = 0, and when x = L/2 then v = ½ v_{e}. *

*Hint: Figure out how to relate dm _{s} to dx and then integrate both sides of the infinitesimal kinetic energy equation to get an equation for the kinetic energy of the spring that includes m_{s}/3. *

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Step 1

*Given,*

Step 2

FBD:

Step 3

Let the *dm* is mass element and *dx* is the...

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