Question

A human hair is about $50\mu m$in diameter. At what speed, in m/s, would a meter stick “shrink by a hair”?

Hint: Use the binomial approximation.

Short answer

At a speed of $3\times {10}^{6}m/s$, a meter stick shrink by a hair.

Step-by-step solution

Given Information

We have given that a human hair is about $50\mu m$in diameter.

We must determine at what pace in m/s a meter stick "shrinks by a hair" using the binomial approximation.

Simplify

$S$: Grounds reference frame

$S\text{'}$: The meter establishes a frame of reference.

$L\text{'}=l(\text{inframe}S\text{'}).$

On the ground, an experimenter measures the length to be constricted to:

$L=\sqrt{1-{\beta }^{2}l}\approx \left(1-\frac{1}{2}{\beta }^2\right)l$

$l-L=\frac{1}{2}{\beta }^{2}l$

As a result, $\beta$has the following value:

$$\begin{gathered} \beta =\sqrt{\frac{2(l-L)}{l}} \\ \beta =\sqrt{\frac{2(50\times {10}^{-6}m)}{1.00m}} \\ \beta =0.01 \end{gathered}$$

Now, we must determine the value of $v$using the following relation:

$$\begin{gathered} v=\beta ·c \\ v=0.01\times (3\times {10}^{8}m/s) \\ v=3\times {10}^{6}m/s \end{gathered}$$