Physics Problems & Solutions: Lab Methods

The magnitude of the force F on an object can be determined by measuring both the mass m of an object and the magnitude of its acceleration a, where F = ma. Assume that these measurements are uncorrelated and normally distributed. if the standard deviations of the measurements of the mass and acceleration are σ_mand σ_arespectively, then σ_F/F is

A. $\left (\frac{\sigma_m}{m} \right )^2+\left (\frac{\sigma_a}{a} \right )^2$

B. $\left (\frac{\sigma_m}{m} +\frac{\sigma_a}{a} \right )^{\frac{1}{2}}$

C. $\left [\left (\frac{\sigma_m}{m} \right )^2+\left (\frac{\sigma_a}{a} \right )^2 \right ]^{\frac{1}{2}}$

D. $\frac{\sigma_m\sigma_a}{ma}$

E. $\frac{\sigma_m}{m}+\frac{\sigma_a}{a}$

(GR9277 #48)

Solution:

Multiplying two quantities with uncertainty:

u = x ⋅ y

$\left ( \frac{\sigma_u}{u} \right )^2=\left (\frac{\sigma_x}{x} \right )^2+\left (\frac{\sigma_y}{y} \right )^2$

For F = ma:

$\frac{\sigma_F}{F}=\left [\left (\frac{\sigma_m}{m} \right )^2+\left (\frac{\sigma_a}{a} \right )^2 \right ]^{\frac{1}{2}}$

Answer: C

Physics Problems & Solutions

Lab Methods - Standard Deviation

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