105k views
5 votes
A certain spring has a spring constant k1​=720N/m as the spring is stretched from x=0 to x1​=39cm. The spring constant then changes to k2​=210N/m as the spring is stretched to x2​=67cm. From x2​=67cm to x3​=91cm the spring force is constant at F3​=115N. Use the area under the curve to calculate the work, in joules, necessary to stretch the spring from x=0 to x1​.

A) 260J

B) 312J

C) 380J

D) 440J

1 Answer

3 votes

Final answer:

The work done to stretch the spring from x=0 to x1=39cm is 104.65J. Therefore, none of the options A, B, C, or D is correct.

Step-by-step explanation:

To calculate the work done to stretch the spring from x=0 to x1=39cm, we need to find the area under the force-extension curve.

The work done by a spring force is given by the formula: W = area under the curve = ∫Fdx

Where

F is the force

dx is the displacement.

First, we calculate the work done from x=0 to x2=67cm using the formula: W = ∫Fdx = ∫115Ndx = 115∫dx.

Since the force is constant and the displacement from x=0 to x2=67cm is 67cm-0cm = 67cm = 0.67m, the work done in this segment is: W = 115N * 0.67m = 77.05J.

Next, we calculate the work done from x2=67cm to x3=91cm. The force in this segment is constant at F3 = 115N.

The displacement is 91cm-67cm = 24cm = 0.24m.

Therefore, the work done in this segment is: W = F3 * dx = 115N * 0.24m = 27.6J.

To find the total work done from x=0 to x1=39cm, we add the work done in each segment: Total work = Work from x=0 to x2 + Work from x2 to x3 = 77.05J + 27.6J = 104.65J.

Rounded to the nearest whole number, the work necessary to stretch the spring from x=0 to x1=39cm is 105J, which is not among the given answer choices.

Therefore, none of the options A, B, C, or D is correct.

User Shmulik
by
8.1k points