In 1999, the Mars Climate Orbiter burned up in the Martian atmosphere. The root cause was a failure to use a uniform system of units. The spacecraft team in Colorado used English units (pound-seconds) for the spacecraft's thruster calculations, while the navigation team in California assumed the data was in SI units (newton-seconds). This mismatch of units led to incorrect trajectory calculations, demonstrating why a standardized system like the SI system is absolutely critical in engineering to ensure clear communication and prevent disastrous failures.

When designing a steel truss bridge, engineers must calculate the forces acting on every joint (node). Wind, traffic, and the bridge's own weight exert forces from various directions. These forces are vector quantities. By resolving these forces into horizontal ($x$) and vertical ($y$) components, engineers can ensure that the sum of forces in all directions equals zero (static equilibrium). Without vector decomposition, determining if a bridge will stand or collapse under stress would be mathematically impossible.

A steel beam is measured to be $15.5$ feet long. Convert this length to meters, given that $1 \text{ inch} = 2.54 \text{ cm}$.

A car on a highway is traveling at $65 \text{ miles/hour}$. Convert this speed to the SI unit of $\text{meters/second}$. ($1 \text{ mile} = 1609 \text{ m}$)

The density of a certain type of concrete is given as $145 \text{ lb/ft}^3$. Convert this density to $\text{kg/m}^3$. ($1 \text{ lb} \approx 0.4536 \text{ kg}$, $1 \text{ ft} = 0.3048 \text{ m}$)

Check if the kinematic equation $v = v_0 + at$ is dimensionally consistent. $v$ and $v_0$ are velocities, $a$ is acceleration, and $t$ is time.

Newton's law of universal gravitation is given by $F = G \frac{m_1 m_2}{r^2}$. Find the dimensions of the gravitational constant $G$.

The period $T$ of a simple pendulum depends on its length $l$ and the acceleration due to gravity $g$. Assuming $T \propto l^x g^y$, use dimensional analysis to find the values of $x$ and $y$.

Determine the number of significant figures in the following measurements: (a) $0.00405$ m, (b) $120.0$ kg, (c) $1500$ s.

Calculate the total mass: $12.4$ kg + $3.15$ kg + $0.082$ kg.

A rectangular steel plate measures $4.52$ cm by $2.1$ cm. It has a mass of $75.5$ g. Calculate its area and area density (mass/area).

A surveyor walks 30 m East, then 40 m North. Find the resultant displacement magnitude and direction.

A force vector $\mathbf{F} = (5\hat{i} + 2\hat{j} - 3\hat{k})$ N acts on an object, moving it by a displacement vector $\mathbf{d} = (3\hat{i} + 4\hat{j} + 1\hat{k})$ m. Calculate the total work done.

A force $\mathbf{F} = (2\hat{i} - 3\hat{j} + 4\hat{k})$ N is applied at a position vector $\mathbf{r} = (3\hat{i} + 2\hat{j} + 1\hat{k})$ m relative to a pivot point. Find the torque vector about the pivot.