Physics 2007

Random errors scatter above and below the true value with no pattern, so their average tends to zero as more readings are taken. Averaging repeated measurements therefore reduces them.

Systematic errors and zero errors shift every reading the same way, so they survive averaging untouched. Parallax is a systematic reading error caused by where the eye is placed.

$x_{\text{true}} = 4.30\ \mathrm{cm}$, $\bar{x} = 4.33\ \mathrm{cm}$, $\sigma = 0.04\ \mathrm{cm}$

Accuracy is measured against the true value; precision is measured by the spread of the readings.

$$\Delta x = |\bar{x} - x_{\text{true}}| = |4.33 - 4.30|\ \mathrm{cm} = 0.03\ \mathrm{cm}$$

The standard deviation $0.04\ \mathrm{cm}$ only says how closely the student's four readings agree with each other, so it is the precision and not the accuracy.

$\vec{A} = a\hat{i} + a\sqrt{3}\hat{j}$, $\vec{B} = a\sqrt{3}\hat{i} + a\hat{j}$

$$\begin{aligned} \vec{A}\cdot\vec{B} &= (a)(a\sqrt{3}) + (a\sqrt{3})(a) = 2\sqrt{3}\,a^2 \\ |\vec{A}| &= \sqrt{a^2 + 3a^2} = 2a \\ |\vec{B}| &= \sqrt{3a^2 + a^2} = 2a \end{aligned}$$

$$\cos\theta = \frac{\vec{A}\cdot\vec{B}}{|\vec{A}||\vec{B}|} = \frac{2\sqrt{3}a^2}{4a^2} = \frac{\sqrt{3}}{2} = 0.866$$

$\theta = 30^\circ$

The resultant is largest when the two vectors point the same way and smallest when they point opposite ways.

$$\begin{aligned} R_{\max} &= 12 + 5 = 17\ \text{units} \\ R_{\min} &= 12 - 5 = 7\ \text{units} \end{aligned}$$

Every possible resultant lies between 7 and 17 units, so 7, 13 and 17 units can all be produced by some angle between the vectors. 60 units lies far outside that range and is impossible.

$\vec{A} = 7\hat{i} + 4\hat{j} - 8\hat{k}$, $\vec{B} = 3\hat{i} - 2\hat{j} + 5\hat{k}$

Take the components one at a time:

$$\begin{aligned} (\vec{A}\times\vec{B})_x &= A_yB_z - A_zB_y = (4)(5) - (-8)(-2) = 20 - 16 = 4 \\ (\vec{A}\times\vec{B})_y &= A_zB_x - A_xB_z = (-8)(3) - (7)(5) = -24 - 35 = -59 \\ (\vec{A}\times\vec{B})_z &= A_xB_y - A_yB_x = (7)(-2) - (4)(3) = -14 - 12 = -26 \end{aligned}$$

$$\vec{A}\times\vec{B} = 4\hat{i} - 59\hat{j} - 26\hat{k}$$

The opposite signs throughout would be $\vec{B}\times\vec{A}$.