find out the final extension and the work

Question

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.