Question

A certain spring has a spring constant k₁ = 420 N/m as the spring is stretched from x = 0 to x₁ = 34 cm. The spring constant then changes to k₂ = 330 N/m as the spring is stretched to x₂ = 79 cm. From x₂ = 79 cm to x₃ = 97 cm the spring force is constant at F₃ = 120 N.

(a) Use the area under the curve to calculate the work, in joules, necessary to stretch the spring from x = 0 to x₁.

(b) Write an equation for the work done in stretching the spring from x₁ to x₂.

(c) Calculate the work done, in joules, in stretching the spring from x₁ to x₂.

(d) Calculate the work, in joules, necessary to stretch the spring from x = 0 to x₃.

Answer 1

Solution-

Consider the force in spring positive, draw the curve as

1. From 0 cm to 34 cm

F = 420 X

2. From 34 cm to 79 cm

F=330 X

3. From 79 cm to 97 cm

F=120

As curve,

(a)

Work was done in 0 to 0.34 m

Calculate area from O to A,

(b)

(c)

Use the force from x=0.34 to 0.79 m

We can use the area of a parallelogram

(d)

The total area will be work done from x=0 to x=0.97 m stretch.

Use AOA and ABC from previous parts,