# determine approximate percentage error using binomial expansion

• Feb 21st 2009, 04:00 AM
decoy808
determine approximate percentage error using binomial expansion
The shear stress τ in a shaft of diameter
D under a torque T is given by:
τ = kT
πD^3 .
Determine the approximate percentage error
in calculating τ if T is measured 3% too
small and D 1.5% too large.

this is a question from J bird engineering mathematics book. it gives the answer but i need to know how to do the method.

any help much appreciated :)
• Feb 21st 2009, 05:19 AM
Approximate increases using binomial expansion
Hello decoy808
Quote:

Originally Posted by decoy808
The shear stress τ in a shaft of diameter
D under a torque T is given by:
τ = kT
πD^3 .
Determine the approximate percentage error
in calculating τ if T is measured 3% too
small and D 1.5% too large.

this is a question from J bird engineering mathematics book. it gives the answer but i need to know how to do the method.

any help much appreciated :)

If $y = kx^n$, and $x$ and $y$ increase by small amounts $\delta x$ and $\delta y$, then

$y+ \delta y = k(x+\delta x)^n$

$\Rightarrow \frac{y+ \delta y}{y }= \frac{k(x+ \delta x)^n}{kx^n}$

$\Rightarrow 1 + \frac{\delta y}{y}= \left(1 + \frac{\delta x}{x}\right)^n$

$\approx 1 + n\frac{\delta x}{x}$, ignoring higher powers of $\frac{\delta x}{x}$, which we may do for small increases in $x$

$\Rightarrow$ the fractional increase in $y \approx n \times$ the fractional increase in $x$.

It is very easy show that this may be extended to a formula for $y$ which involves powers of any number of variables.

So, using the same method you may prove that if $\tau = \frac{k}{\pi}T\cdot D^{-3}$, then

$\frac{\delta \tau}{\tau} \approx \frac{\delta T}{T} -3 \frac{\delta D}{D}$

With the percentage errors you have been given

$\frac{\delta T}{T} = -0.03$ and $\frac{\delta D}{D} = 0.015$

So $\frac{\delta \tau}{\tau} \approx -0.03 -0.045 = -0.075 = -7.5$%