Find out the final extension and the work done, Mechanical Engineering

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Find out the final extension and the work done:

Problem:

 (i) A spring of stiffness 4 N/cm of extension is extended steadily yet the force of 20 N is reached. Find out the final extension and the work done to cause this extension.

(ii) If the extension is further to be enhanced by 20%, how much additional work is necessary. What is the ultimate magnitude of force needed to achieve this.

Solution

Stiffness (K) of a spring is described as force required to cause unit deflection. Therefore, if s1 is the deflection under a force F1

∴ Stiffness = F1/ s1 = tan θ

∴ 4 = 20 /s1

or,

s1 = 20 /4= 5 cm

 (i)        W. D. on spring to cause extension s1 = Area of Δ Os1 A

= (20 × 5) /2 = 50 N-cm.

Further 20% extension beyond s1 = 0.2 × 5 = 1 cm

s1  s2  = 1 cm

s2 = Total Extension = Os2 = 6 cm

∴ F2 = Stiffness × 6 = 24 N.

 (ii)       Additional WD from s1 to s2 = Area of trapezium As1 s2 B

=( (F1 + F2 )/2 ) × s1  s2

= ((20 + 24)/2) × 1= 22 N-cm.


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