Progressive Measurement and Vector Examples

A steel tape gives a beam length of $15.5 \text{ ft}$. Convert the measurement to meters using $1 \text{ ft} = 12 \text{ in}$, $1 \text{ in} = 2.54 \text{ cm}$, and $100 \text{ cm} = 1 \text{ m}$.

Normal-weight concrete is listed as $145. \text{ lb/ft}^3$. Convert the density to $\text{kg/m}^3$ using $1 \text{ lb} = 0.453592 \text{ kg}$ and $1 \text{ ft} = 0.3048 \text{ m}$.

A cart with mass $12.0 \text{ kg}$ accelerates at $2.50 \text{ m/s}^2$. Find the force in newtons and show the equivalent base SI units.

Verify that the work relationship $W = Fd$ is dimensionally consistent when $F$ is force and $d$ is displacement.

Determine the number of significant figures in these measurements: (a) $0.00405 \text{ m}$, (b) $120.0 \text{ kg}$, and (c) $1.500 \times 10^3 \text{ s}$.

A rectangular steel plate measures $4.52 \text{ cm}$ by $2.1 \text{ cm}$ and has a mass of $75.5 \text{ g}$. Calculate its area and area density.

A survey rod segment with accepted length $2.000 \text{ m}$ is measured as $1.998 \text{ m}$. Calculate the percent error of the measurement.

A force of $250 \text{ N}$ acts at $35.0^\circ$ above the positive $x$-axis. Find its $x$- and $y$-components.

Two forces act at a joint: $\vec{F}_1 = 120 \text{ N}$ due east and $\vec{F}_2 = 80.0 \text{ N}$ at $30.0^\circ$ north of east. Find the resultant magnitude and direction.

A displacement vector is $\vec{d} = (-6\hat{i} + 8\hat{j} + 3\hat{k}) \text{ m}$. Find its magnitude and write the corresponding unit direction vector.

A force $\vec{F} = (5\hat{i} + 2\hat{j} - 3\hat{k}) \text{ N}$ moves an object through $\vec{d} = (3\hat{i} + 4\hat{j} + 1\hat{k}) \text{ m}$. Calculate the work done.

A force $\vec{F} = (2\hat{i} - 3\hat{j} + 4\hat{k}) \text{ N}$ is applied at position $\vec{r} = (3\hat{i} + 2\hat{j} + 1\hat{k}) \text{ m}$ from a pivot. Find the torque $\vec{\tau} = \vec{r} \times \vec{F}$.