This is frankly more of a math exercise than a physics one, but it is good pract

Question

This is frankly more of a math exercise than a physics one, but it is good practice in how physicists use approximations. On page 12 of the notes from lecture 13 (slidesl13 update.pdf on canvas under Files May 1 Lecture Materials), l state that AV in other words that if, for example, volume goes up by 1%, density goes down by 1%. This is only approximate, and the approximation is only true for small changes. Its also used in section 16-2 of but not really examined. A) Argue that it must be approximate by discussing a case where the volume is reduced by 95% to 0.05 times its original value, and seeing if what happens to the density makes sense. B) Write the exact version of the formula for App as a function of AV and V by starting from the notion that the mass, pV, doesnt change when p increases by Ap and V decreases by AV Show AV that it can be written in the form Ap V +AV Now go back to the case in A) and show it makes sense with the fixed formula. C) Multiply out the product (p+Ap)(V+AV), and throw out the term that is doubly small (product of two deltas) when each delta is small. The product of two small numbers can often be neglected relative to the other terms. From that point in the derivation in part B), now that you have made the approximation, go back and recover the formula that starts this problem.

Explanation / Answer

A) In case of 95% change in volume (reduction)
d(rho)/rho = -d(V)/V = -0.95
now, rho*V = mass = constant
rho = constant/V
rho - d(rho) = constant/(V + dV)
but constant = rho*V
rho - d(rho) = rho*V/(V +dV)
1 - d(rho)/rho = V/(V + dV)
d(rho)/rho = -dV/(V + dV)
so, rho/d(rho) = -(V/dV + 1)
rho/d(rho) = -(-1/.95 + 1) = 00.05263 = 5.263 pc
so, d(rho)/rho = 19 pc
hence the given equation is just an apprximation
B) From A< d(rho)/rho = -dV/(V + dV)
C) d(rho)*(rho + d(rho))(V + dV)/rho = -dV*(rho*d(rho))
[d(rho)*rho + d(rho)*d(rho)](V + dV)/rho = -dV*rho*d(rho)
ignoring double productis
d(rho)*rho*(V + dV)/rho = -dV*rho*d(rho)
d(rho)/rho = -dV/V

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