A student says "percentage errors always add." For which operation is this statement correct? <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 400 140" style="max-width:400px;width:100%;height:auto;background:#faf9f5;border-radius:12px;border:1px solid rgba(20,20,19,0.10)"> <title>Error rules summary — absolute vs relative errors for different operations</title> <!-- Sum/Difference box --> <rect x="15" y="15" width="175" height="55" rx="8" fill="#c6613f" opacity="0.08" stroke="#c6613f" stroke-width="1.5"/> <text x="103" y="33" text-anchor="middle" font-size="11" font-weight="700" fill="#c6613f">A + B or A - B</text> <text x="103" y="50" text-anchor="middle" font-size="11" fill="#14140f">Absolute errors add</text> <text x="103" y="64" text-anchor="middle" font-size="10" fill="#14140f">$\Delta$(A±B) = $\Delta A$ + $\Delta B$</text> <!-- Product/Quotient box --> <rect x="210" y="15" width="175" height="55" rx="8" fill="#788c5d" opacity="0.12" stroke="#788c5d" stroke-width="1.5"/> <text x="298" y="33" text-anchor="middle" font-size="11" font-weight="700" fill="#788c5d">A × B or A / B</text> <text x="298" y="50" text-anchor="middle" font-size="11" fill="#14140f">Relative errors add</text> <text x="298" y="64" text-anchor="middle" font-size="10" fill="#14140f">%err = %err(A) + %err(B)</text> <!-- Power box --> <rect x="110" y="85" width="180" height="40" rx="8" fill="#14140f" opacity="0.04" stroke="#14140f" stroke-width="1"/> <text x="200" y="102" text-anchor="middle" font-size="11" font-weight="700" fill="#14140f">Aⁿ</text> <text x="200" y="118" text-anchor="middle" font-size="11" fill="#14140f">%err(Aⁿ) = n × %err(A)</text> </svg>

The equation v = sqrt(2gh) is dimensionally correct. Does this guarantee it is physically correct?

A student claims "since work and torque have the same dimensions, they are the same physical quantity." This is:

In y = A*sin(omega*t + phi), which of the following must be dimensionless?

Two masses are measured: m_1 = 100.0 +/- 0.5 g and m_2 = 99.0 +/- 0.5 g. The percentage error in (m_1 - m_2) is: <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 400 140" style="max-width:400px;width:100%;height:auto;background:#faf9f5;border-radius:12px;border:1px solid rgba(20,20,19,0.10)"> <title>Subtraction error amplification — nearly equal values give enormous % error</title> <!-- m1 bar --> <rect x="30" y="25" width="300" height="20" rx="2" fill="#14140f" opacity="0.08" stroke="#14140f" stroke-width="1"/> <text x="340" y="40" font-size="10" fill="#14140f">m_{1} = 100.0 ± 0.5 g</text> <!-- m2 bar --> <rect x="30" y="50" width="297" height="20" rx="2" fill="#788c5d" opacity="0.12" stroke="#788c5d" stroke-width="1"/> <text x="340" y="65" font-size="10" fill="#788c5d">m_{2} = 99.0 ± 0.5 g</text> <!-- Difference highlight --> <rect x="327" y="25" width="3" height="45" fill="#c6613f"/> <text x="335" y="52" font-size="9" font-weight="700" fill="#c6613f">1.0g</text> <!-- Error calculation --> <text x="200" y="95" text-anchor="middle" font-size="11" fill="#14140f">$\Delta$(m_{1}−m_{2}) = 0.5 + 0.5 = 1.0 g</text> <text x="200" y="115" text-anchor="middle" font-size="14" font-weight="700" fill="#c6613f">% error = 1.0/1.0 × 100 = 100%</text> <text x="200" y="135" text-anchor="middle" font-size="10" fill="#14140f">Individual: 0.5% each → Difference: 100%!</text> </svg>