Q. 4.31 - Elementary Statistics

Question

Hooke's Law. According to Hooke's law for springs, developed by Robert Hooke ($1635-1703$), the force exerted by a spring that has been compressed to a length $x$ is given by the formula $F=-k\left(x-x_0\right)$, where $x_0$ is the natural length of the spring and $k$ is a constant, called the spring constant. A certain spring exerts a force of $32 lb$ when compressed to a length of $2 ft$ and a force of $16 lb$ when compressed to a length of $3 ft$. For this spring, find the following

a. The linear equation that relates the force exerted to the length compressed

b. The spring constant

c. The natural length of the spring

Step-by-Step Solution

Answer

(a) The linear equation is, $F=16\left(x-4\right)$

(b) The spring constant is, $k=16$

(c) The natural length is, $x_0=4$

Part (a) Step 1: Given information

We need to find out the corresponding linear equation.

Part (a) Step 2: Simplify

The equation is, $F=-k(x-x_0)$

Now,

If $x=2$, then $F=32$

Therefore,

         $$\begin{gathered} 32=-k(2-x_0) \\ k=\frac{32}{x_0-2} \end{gathered}$$

Similarly if $x=3$ then $F=16$

Therefore, $16=-k\left(3-x_0\right)$

As the value of $k$ is given above

So, the equation becomes, 

    $$\begin{gathered} 16=-\frac{32}{x_0-2}\times \left(3-x_0\right) \\ 16\left(x_0-2\right)=-32\left(3-x_0\right) \\ 16x_0-32=32x_0-96 \\ 16x_0=64 \\ {x}_0=4 \end{gathered}$$

Now, 

    $k=\frac{32}{x_0-2}=\frac{32}{4-2}=16$

Hence, The required linear equation  is, 

 $$\begin{gathered} F=k(x-x_0) \\ F=16(x-4) \end{gathered}$$

Part (a) Step 3: Explanation

Here, $F$ is the force experienced by spring

And $x$ is the length of spring compressed

And $x_0$ is the natural length of spring

And $k$ is spring constant

Part (b) Step 1: Given information

We need to find the value of spring constant

Part (b) Step 2: Simplify

From part (a)

The value of the spring constant is,

   $k=16$

Part (c) Step 1: Given information

We need to find out the value of natural length 

Part (c) Step 2: Simplify

From part (a),

The value of natural length is,

    $x_0=4$