Question

A bungee cord is essentially a very long rubber band that can stretch up to four times its unstretched length. However, its spring constant varies over its stretch. Take the length of the cord to be along the x-direction and define the stretch x as the length of the cord l minus its un-stretched length
`l_0`; that is,
`x=l-l_0` (see below). Suppose a particular bungee cord has a spring constant, for 0 ≤ x ≤ 4.88m , of
`k_1=204 N/m` and for x ≥ 4.88m , of
`k_2=111N/m`. (Recall that the spring constant is the slope of the force F(x) versus its stretch x.) (a) What is the tension in the cord when the stretch is 16.7 m (the maximum desired for a given jump)? (b) How much work must be done against the elastic force of the bungee cord to stretch it 16.7 m? please explain if possible!

Answer 1

(a) To find the tension in the cord when the stretch is 16.7 m, we need to first determine which spring constant applies to this stretch. Since 16.7 m is greater than 4.88 m, the spring constant for x ≥ 4.88 m applies, which is

�

2

=

111

�

/

�

k

2

=111N/m.

Next, we need to find the force exerted by the bungee cord at this stretch. The force F(x) exerted by a spring is given by:

F(x) = kx

where k is the spring constant and x is the stretch. Plugging in the values for k and x, we get:

F(16.7) = (111 N/m)(16.7 m) = 1853.7 N

Therefore, the tension in the cord when the stretch is 16.7 m is 1853.7 N.

(b) To find the work done against the elastic force of the bungee cord to stretch it 16.7 m, we need to integrate the force over the stretch. Since the spring constant changes at 4.88 m, we need to break up the integration into two parts.

For 0 ≤ x ≤ 4.88 m, the force is given by:

F(x) = k

1

x

where k

1

= 204 N/m. Integrating this expression over the stretch from 0 to 4.88 m, we get:

W

1

= ∫

4.88

0

k

1

x dx = (204 N/m) * (4.88 m)

2

/ 2 = 996.8 J

For 4.88 m ≤ x ≤ 16.7 m, the force is given by:

F(x) = k

2

x

where k

2

= 111 N/m. Integrating this expression over the stretch from 4.88 m to 16.7 m, we get:

W

2

= ∫

16.7

4.88

k

2

x dx = (111 N/m) * (16.7 m)

2

/ 2 - (111 N/m) * (4.88 m)

2

/ 2 = 1232.8 J

Therefore, the total work done against the elastic force of the bungee cord to stretch it 16.7 m is:

W = W

1

+ W

2

= 996.8 J + 1232.8 J = 2229.6 J