- #1
Lotto
- 231
- 14
- Homework Statement
- Round ##v_0=\sqrt{5gl-\frac{1}{4\pi \epsilon_0}\frac{Q^2}{ml}}## correctly.
##Q =1.0 \, \mathrm{\mu C}##, ##l=10 \,\mathrm {cm}##, ##m = 150 \,\mathrm g##, ##\epsilon_0=8.85 \cdot 10^{-12} \, \mathrm {F\cdot m^{-1}}##, ##g=9.81\,\mathrm{m\cdot s^{-2}}##.
- Relevant Equations
- ##v_0=\sqrt{5gl-\frac{1}{4\pi \epsilon_0}\frac{Q^2}{ml}}##
I have never been sure how to round it according to the rules. My steps:
##\sqrt{5 \cdot 9.81 \cdot 0.10-\frac{1}{4\cdot \pi \cdot 8.85 \cdot 10^{-12}}\frac{(1.0\cdot 10^{-6})^2}{0.150\cdot 0.10}}##
In [ ] is number of significant digits using multiplication/adding etc. rules.
##\sqrt{5 \cdot 9.81 \cdot 0.10-\frac{1}{4\cdot \pi \cdot 8.85 \cdot 10^{-12}}\frac{(1.0\cdot 10^{-6})^2}{0.150\cdot 0.10}}##
##\sqrt{4.905[2]-8991804694[3]\cdot \frac{1.0[2]\cdot 10^{-12}}{0.0150[2]}}##
##\sqrt{4.905[2]-8991804694[3]\cdot 6.666667[2]\cdot 10^{-11}}##
##\sqrt{4.908[2]-0.59945[2]}##
##\sqrt{4.30855[2]}##
##v_0=2.1 \,\mathrm{m\cdot s^{-1}}.##
Is it correct according to the rules? But this is pain to calculate. I would normally put it all into a calculator and write the result on one decimal place, without using any rules.
How to round in calculations correctly and effieciently?
##\sqrt{5 \cdot 9.81 \cdot 0.10-\frac{1}{4\cdot \pi \cdot 8.85 \cdot 10^{-12}}\frac{(1.0\cdot 10^{-6})^2}{0.150\cdot 0.10}}##
In [ ] is number of significant digits using multiplication/adding etc. rules.
##\sqrt{5 \cdot 9.81 \cdot 0.10-\frac{1}{4\cdot \pi \cdot 8.85 \cdot 10^{-12}}\frac{(1.0\cdot 10^{-6})^2}{0.150\cdot 0.10}}##
##\sqrt{4.905[2]-8991804694[3]\cdot \frac{1.0[2]\cdot 10^{-12}}{0.0150[2]}}##
##\sqrt{4.905[2]-8991804694[3]\cdot 6.666667[2]\cdot 10^{-11}}##
##\sqrt{4.908[2]-0.59945[2]}##
##\sqrt{4.30855[2]}##
##v_0=2.1 \,\mathrm{m\cdot s^{-1}}.##
Is it correct according to the rules? But this is pain to calculate. I would normally put it all into a calculator and write the result on one decimal place, without using any rules.
How to round in calculations correctly and effieciently?
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