document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1181818>Question 1181818</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A rectangular brass bar of mass M has dimensions a, b, c as shown in the<BR>\n" );
document.write( "figure below. The moment of inertia I about the axis at the centre of the ab<BR>\n" );
document.write( "face and perpendicular to it, is given by:\r<BR>\n" );
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document.write( "I=(M/12)(a^2 + b^2)\r<BR>\n" );
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document.write( "The following measurements are made:<BR>\n" );
document.write( "𝑀=135.0±0.1𝑔<BR>\n" );
document.write( "𝑎 =80±1 𝑚𝑚<BR>\n" );
document.write( "𝑏 =10±1 𝑚𝑚<BR>\n" );
document.write( "𝑐 =20.00±0.01 𝑚𝑚\r<BR>\n" );
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document.write( "Calculate the standard error in:<BR>\n" );
document.write( "a) the density ρ of the material<BR>\n" );
document.write( "b) the moment of inertia I </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849962 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849962\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to calculate the standard error in density (ρ) and moment of inertia (I):\r<BR>\n" );
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document.write( "**a) Standard Error in Density (ρ)**\r<BR>\n" );
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document.write( "1. **Calculate the nominal density:**<BR>\n" );
document.write( "   ρ = M / (abc) = 135.0 g / (80 mm * 10 mm * 20.00 mm) = 0.084375 g/mm³\r<BR>\n" );
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document.write( "2. **Calculate the fractional uncertainties:**<BR>\n" );
document.write( "   * ΔM/M = 0.1 g / 135.0 g = 0.000741<BR>\n" );
document.write( "   * Δa/a = 1 mm / 80 mm = 0.0125<BR>\n" );
document.write( "   * Δb/b = 1 mm / 10 mm = 0.1<BR>\n" );
document.write( "   * Δc/c = 0.01 mm / 20.00 mm = 0.0005\r<BR>\n" );
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document.write( "3. **Calculate the fractional uncertainty in density:**<BR>\n" );
document.write( "   (Δρ/ρ)² = (ΔM/M)² + (Δa/a)² + (Δb/b)² + (Δc/c)²<BR>\n" );
document.write( "   (Δρ/ρ)² = (0.000741)² + (0.0125)² + (0.1)² + (0.0005)²<BR>\n" );
document.write( "   (Δρ/ρ)² = 0.00000055 + 0.00015625 + 0.01 + 0.00000025 = 0.010157<BR>\n" );
document.write( "   Δρ/ρ = √0.010157 ≈ 0.1008\r<BR>\n" );
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document.write( "4. **Calculate the absolute uncertainty in density:**<BR>\n" );
document.write( "   Δρ = (Δρ/ρ) * ρ = 0.1008 * 0.084375 g/mm³ ≈ 0.0085 g/mm³\r<BR>\n" );
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document.write( "5. **Express the density with its uncertainty:**<BR>\n" );
document.write( "   ρ = 0.0844 ± 0.0085 g/mm³ (rounded to appropriate significant figures)\r<BR>\n" );
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document.write( "**b) Standard Error in Moment of Inertia (I)**\r<BR>\n" );
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document.write( "1. **Calculate the nominal moment of inertia:**<BR>\n" );
document.write( "   I = (M/12)(a² + b²) = (135.0 g / 12) * ((80 mm)² + (10 mm)²) = 11.25 g * (6400 mm² + 100 mm²) = 11.25 g * 6500 mm² = 73125 g mm²\r<BR>\n" );
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document.write( "2. **Calculate the fractional uncertainties:** (Same as in part a)<BR>\n" );
document.write( "   * ΔM/M = 0.000741<BR>\n" );
document.write( "   * Δa/a = 0.0125<BR>\n" );
document.write( "   * Δb/b = 0.1\r<BR>\n" );
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document.write( "3. **Calculate the fractional uncertainty in I:**<BR>\n" );
document.write( "  (ΔI/I)² = (ΔM/M)² + [2*(Δa/a)]² + [2*(Δb/b)]²<BR>\n" );
document.write( "  (ΔI/I)² = (0.000741)² + [2*(0.0125)]² + [2*(0.1)]²<BR>\n" );
document.write( " (ΔI/I)² = 0.00000055 + 0.000625 + 0.04 = 0.04062555<BR>\n" );
document.write( "  ΔI/I = √0.04062555 ≈ 0.2016\r<BR>\n" );
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document.write( "4. **Calculate the absolute uncertainty in I:**<BR>\n" );
document.write( "   ΔI = (ΔI/I) * I = 0.2016 * 73125 g mm² ≈ 14750 g mm²\r<BR>\n" );
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document.write( "5. **Express the moment of inertia with its uncertainty:**<BR>\n" );
document.write( "   I = 73100 ± 14800 g mm² (rounded to appropriate significant figures)\r<BR>\n" );
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document.write( "**Key Points:**\r<BR>\n" );
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document.write( "*   The uncertainty in *b* has the largest impact on both ρ and I because it has the largest fractional uncertainty.<BR>\n" );
document.write( "*   When calculating the uncertainty in I, the uncertainties in *a* and *b* are multiplied by 2 because they are squared in the formula for I.<BR>\n" );
document.write( "*   Always round your final answers to the appropriate number of significant figures.  The uncertainties should usually have only one significant figure (sometimes two if the leading digit is a 1 or 2), and the measured values should be rounded to the same decimal place as the uncertainty.<BR>\n" );
document.write( " </SPAN>\n" );
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