If a couplo mgR is applled at the end of a wire, the angle of twist \( \theta \) at a dist from tho fixed end is given by \[ \mathrm{mgR}=\frac{n a \theta}{21} \] Calculate \( n \) and its standard error given the following values: \( 0=0.200 \pm 0.009 \mathrm{rad} ; \mathrm{m}=3.500 \mathrm{~kg} \mathrm{I}=0.330 \pm 0.002 \mathrm{~m} \); \( 1=0.0047 \pm 0.0001 \mathrm{~m} ; \mathrm{R}=0.074 \pm 0.001 \mathrm{~m} ; g=9.760 \mathrm{~ms}^{-2} \) (Assume that the ( m end g are negligible)

To calculate \( n \) and its standard error, we first need to rearrange the given equation to solve for \( n \). Given equation: \[ \mathrm{mgR}=\frac{n a \theta}{21} \] Rearrange the equation to solve for \( n \): \[ n = \frac{21 \times \mathrm{mgR}}{a \theta} \] Now, we can substitute the given values into the equation to calculate \( n \) and its standard error. Given values: - \( \theta = 0.200 \pm 0.009 \) rad - \( m = 3.500 \) kg - \( g = 9.760 \) m/s\(^2\) - \( R = 0.074 \pm 0.001 \) m - \( a = 0.330 \pm 0.002 \) m - \( I = 0.0047 \pm 0.0001 \) m Substitute the values into the equation to calculate \( n \): \[ n = \frac{21 \times 3.500 \times 9.760 \times 0.074}{0.330 \times 0.200} \] Calculate \( n \) using the given values. Calculate the value by following steps: - step0: Calculate: \(\frac{\frac{21\times 3.5\times 9.76\times 0.074}{0.33}}{0.2}\) - step1: Multiply the terms: \(\frac{\frac{53.08464}{0.33}}{0.2}\) - step2: Divide the terms: \(\frac{\frac{221186}{1375}}{0.2}\) - step3: Convert the expressions: \(\frac{\frac{221186}{1375}}{\frac{1}{5}}\) - step4: Multiply by the reciprocal: \(\frac{221186}{1375}\times 5\) - step5: Reduce the numbers: \(\frac{221186}{275}\times 1\) - step6: Multiply: \(\frac{221186}{275}\) The value of \( n \) is approximately 804.31. To calculate the standard error of \( n \), we need to consider the standard errors of the given values and propagate them through the calculation. The standard error of \( n \) can be calculated using the formula: \[ \text{Standard Error of } n = \sqrt{\left(\frac{\partial n}{\partial \theta}\right)^2 \cdot \text{Standard Error of } \theta + \left(\frac{\partial n}{\partial a}\right)^2 \cdot \text{Standard Error of } a + \left(\frac{\partial n}{\partial R}\right)^2 \cdot \text{Standard Error of } R} \] Let's calculate the standard error of \( n \) using the given values. Calculate the value by following steps: - step0: Calculate: \(\sqrt{\left(\frac{\frac{21\times 3.5\times 9.76\times 0.074}{0.33}}{0.2}\right)^{2}\times 0.009^{2}+\left(\frac{\frac{21\times 3.5\times 9.76\times 0.074}{0.33}}{0.2}\right)^{2}\times 0.002^{2}+\left(\frac{\frac{21\times 3.5\times 9.76\times 0.074}{0.33}}{0.2}\right)^{2}\times 0.001^{2}}\) - step1: Multiply the terms: \(\sqrt{\left(\frac{\frac{53.08464}{0.33}}{0.2}\right)^{2}\times 0.009^{2}+\left(\frac{\frac{21\times 3.5\times 9.76\times 0.074}{0.33}}{0.2}\right)^{2}\times 0.002^{2}+\left(\frac{\frac{21\times 3.5\times 9.76\times 0.074}{0.33}}{0.2}\right)^{2}\times 0.001^{2}}\) - step2: Divide the terms: \(\sqrt{\left(\frac{\frac{221186}{1375}}{0.2}\right)^{2}\times 0.009^{2}+\left(\frac{\frac{21\times 3.5\times 9.76\times 0.074}{0.33}}{0.2}\right)^{2}\times 0.002^{2}+\left(\frac{\frac{21\times 3.5\times 9.76\times 0.074}{0.33}}{0.2}\right)^{2}\times 0.001^{2}}\) - step3: Divide the numbers: \(\sqrt{\left(\frac{221186}{275}\right)^{2}\times 0.009^{2}+\left(\frac{\frac{21\times 3.5\times 9.76\times 0.074}{0.33}}{0.2}\right)^{2}\times 0.002^{2}+\left(\frac{\frac{21\times 3.5\times 9.76\times 0.074}{0.33}}{0.2}\right)^{2}\times 0.001^{2}}\) - step4: Multiply the terms: \(\sqrt{\left(\frac{221186}{275}\right)^{2}\times 0.009^{2}+\left(\frac{\frac{53.08464}{0.33}}{0.2}\right)^{2}\times 0.002^{2}+\left(\frac{\frac{21\times 3.5\times 9.76\times 0.074}{0.33}}{0.2}\right)^{2}\times 0.001^{2}}\) - step5: Divide the terms: \(\sqrt{\left(\frac{221186}{275}\right)^{2}\times 0.009^{2}+\left(\frac{\frac{221186}{1375}}{0.2}\right)^{2}\times 0.002^{2}+\left(\frac{\frac{21\times 3.5\times 9.76\times 0.074}{0.33}}{0.2}\right)^{2}\times 0.001^{2}}\) - step6: Divide the numbers: \(\sqrt{\left(\frac{221186}{275}\right)^{2}\times 0.009^{2}+\left(\frac{221186}{275}\right)^{2}\times 0.002^{2}+\left(\frac{\frac{21\times 3.5\times 9.76\times 0.074}{0.33}}{0.2}\right)^{2}\times 0.001^{2}}\) - step7: Multiply the terms: \(\sqrt{\left(\frac{221186}{275}\right)^{2}\times 0.009^{2}+\left(\frac{221186}{275}\right)^{2}\times 0.002^{2}+\left(\frac{\frac{53.08464}{0.33}}{0.2}\right)^{2}\times 0.001^{2}}\) - step8: Divide the terms: \(\sqrt{\left(\frac{221186}{275}\right)^{2}\times 0.009^{2}+\left(\frac{221186}{275}\right)^{2}\times 0.002^{2}+\left(\frac{\frac{221186}{1375}}{0.2}\right)^{2}\times 0.001^{2}}\) - step9: Divide the numbers: \(\sqrt{\left(\frac{221186}{275}\right)^{2}\times 0.009^{2}+\left(\frac{221186}{275}\right)^{2}\times 0.002^{2}+\left(\frac{221186}{275}\right)^{2}\times 0.001^{2}}\) - step10: Convert the expressions: \(\sqrt{\left(\frac{221186}{275}\right)^{2}\left(\frac{9}{1000}\right)^{2}+\left(\frac{221186}{275}\right)^{2}\times 0.002^{2}+\left(\frac{221186}{275}\right)^{2}\times 0.001^{2}}\) - step11: Convert the expressions: \(\sqrt{\left(\frac{221186}{275}\right)^{2}\left(\frac{9}{1000}\right)^{2}+\left(\frac{221186}{275}\right)^{2}\left(\frac{1}{500}\right)^{2}+\left(\frac{221186}{275}\right)^{2}\times 0.001^{2}}\) - step12: Convert the expressions: \(\sqrt{\left(\frac{221186}{275}\right)^{2}\left(\frac{9}{1000}\right)^{2}+\left(\frac{221186}{275}\right)^{2}\left(\frac{1}{500}\right)^{2}+\left(\frac{221186}{275}\right)^{2}\left(\frac{1}{1000}\right)^{2}}\) - step13: Multiply the numbers: \(\sqrt{\frac{995337^{2}}{137500^{2}}+\left(\frac{221186}{275}\right)^{2}\left(\frac{1}{500}\right)^{2}+\left(\frac{221186}{275}\right)^{2}\left(\frac{1}{1000}\right)^{2}}\) - step14: Multiply the numbers: \(\sqrt{\frac{995337^{2}}{137500^{2}}+\frac{110593^{2}}{68750^{2}}+\left(\frac{221186}{275}\right)^{2}\left(\frac{1}{1000}\right)^{2}}\) - step15: Multiply the numbers: \(\sqrt{\frac{995337^{2}}{137500^{2}}+\frac{110593^{2}}{68750^{2}}+\frac{110593^{2}}{137500^{2}}}\) - step16: Add the numbers: \(\sqrt{\frac{995337^{2}+221186^{2}+110593^{2}}{137500^{2}}}\) - step17: Use the properties of radicals: \(\frac{\sqrt{995337^{2}+221186^{2}+110593^{2}}}{\sqrt{137500^{2}}}\) - step18: Simplify the expression: \(\frac{110593\sqrt{81+4+1}}{137500}\) - step19: Add the numbers: \(\frac{110593\sqrt{86}}{137500}\) The standard error of \( n \) is approximately 7.46. Therefore, the value of \( n \) is approximately 804.31 with a standard error of approximately 7.46.