Question

a physical quantity is expressed as x=ab.If delta a,delta b are absolute errors in measurement of a,b respectively then deduce an expression to determine absolute error in product x​

Answer 1

Given info : a physical quantity is expressed as x = ab , here ∆a , ∆b are abaolute errors in measurements of a and b respectively.

We have to deduce the expression to determine absolute error in product x.

Solution : here equation is, x = ab

Taking log base e both sides,

lnx = ln(ab)

⇒lnx = lna + lnb [ from logarithmic property ]

Now differentiating both sides

⇒d(lnx) = d(lna + lnb) = d(lna) + d(lnb)

⇒dx/x = da/a + db/b

If error is not infinitesimally small.

Then, dx = ∆x , da = ∆a and db = ∆b

so, ∆x/x = ∆a/a + ∆b/b

⇒∆x = x(∆a/a + ∆b/b)

⇒∆x = ab(∆a/a + ∆b/b) = b∆a + a∆b

Therefore the expression of absolute error in measurement of x is given by, ∆x = b∆a + a∆b