Question Number 220221 by fantastic last updated on 08/May/25
$${Q}.{The}\:{density}\:{of}\:{an}\:{object}\:{of}\:{mass}\:{M}\:{is}\:\delta\:{and}\:{the}\:{density}\:{of}\:{the}\:{air}\:{is}\:\rho. \\ $$$${the}\:{mass}\:{of}\:{of}\:{the}\:{object}\:{is}\:{measured}\:{with}\:\:{the}\:{help}\:{of}\:{a}\:{metal}\:{weight}\:{of}\:{mass}\:{m}\:. \\ $$$${the}\:{density}\:{of}\:{the}\:{metal}\:{weight}\:{is}\:{d}. \\ $$$${if}\:\rho\ll\delta\:{them}\:{show}\:{that}\:{the}\:{real}\:{mass}\:{M}\:{will}\:{be} \\ $$$${m}\left(\mathrm{1}−\frac{\rho}{{d}}\:\right)\left(\mathrm{1}+\frac{\rho}{\delta}\right) \\ $$$${I}\:{have}\:{managed}\:{to}\:{M}=\frac{{m}\left(\mathrm{1}−\frac{\rho}{{d}}\right)}{\left(\mathrm{1}−\frac{\rho}{\delta}\right)} \\ $$$${but}\:{I}\:{can}\:{not}\:{figure}\:{it}\:{to}\:{the}\:{end} \\ $$$${please}\:{help} \\ $$
Commented by mr W last updated on 09/May/25
$$\frac{\mathrm{1}}{\mathrm{1}−{x}}=\mathrm{1}+{x}+{x}^{\mathrm{2}} +{x}^{\mathrm{3}} +…\:\:\:\:\:\:\:\left(\mid{x}\mid<\mathrm{1}\right) \\ $$$${if}\:\mid{x}\mid\ll\mathrm{1}: \\ $$$$\:{x}^{\mathrm{2}} \rightarrow\mathrm{0},\:{x}^{\mathrm{3}} \rightarrow\mathrm{0},\:{etc}. \\ $$$$\Rightarrow\:\frac{\mathrm{1}}{\mathrm{1}−{x}}\approx\mathrm{1}+{x} \\ $$$${since}\:\rho\ll\delta,\:{i}.{e}.\:\frac{\rho}{\delta}\ll\mathrm{1} \\ $$$$\Rightarrow\frac{\mathrm{1}}{\mathrm{1}−\frac{\rho}{\delta}}\approx\mathrm{1}+\frac{\rho}{\delta} \\ $$
Commented by fantastic last updated on 09/May/25
$${thanks} \\ $$