Question

Question: An archer pulls her bowstring back 0.4m by exerting a force that increases uniformly from zero to 230N.

(a) What is the equivalent spring constant of the bow?

(b) How much work does the archer do on the string in drawing the bow?

Short answer

(a) The equivalent spring constant of the bow is $575N/m$.

(b) The work done by the archer on the string in drawing the bow is $46J$.

Step-by-step solution

Given data

Stretch of the bow

$x=0.4\mathrm{m}$

Maximum force on the bow

$F=230\mathrm{N}$

Spring force and work done on a spring

The force that tries to bring a spring to its equilibrium position when it is extended by distance x is

$\mathrm{F}=-\mathrm{kx}$ .....(I)

Here, k is the spring constant.

The work done by a force on a spring of spring constant k when it is extended by x is

$W=\frac{1}{2}k{x}^2$ .....(II)

Determining the equivalent spring constant of the bow

From equation (I), the equivalent spring constant of the bow is

$\begin{array}{c}k=\frac{F}{x}\\ =\frac{230\mathrm{N}}{0.4\mathrm{m}}\\ =575\mathrm{N}/\mathrm{m}\end{array}$

Thus, the required spring constant is $575\mathrm{N}/\mathrm{m}$.

Determining the work done on the bow

From equation (II), the work done on the bow is

$\begin{array}{c}W=\frac{1}{2}\times 575\mathrm{N}/\mathrm{m}\times {\left(0.4\mathrm{m}\right)}^2\\ =46·\left(1\mathrm{N}·\mathrm{m}\times \frac{1\mathrm{J}}{1\mathrm{N}·\mathrm{m}}\right)\\ =46\mathrm{J}\end{array}$

Thus, the required work done is $46\mathrm{J}$.