Two solid uniform spheres are placed $2.500 \text{ m}$ apart, measured from center to center. Sphere A has a mass of $400.0 \text{ kg}$, and Sphere B has a mass of $150.0 \text{ kg}$. Determine the gravitational force of attraction between the two spheres. (Use $G = 66.73 \times 10^{-12} \text{ m}^3/(\text{kg}\cdot\text{s}^2)$)

A steel beam has a mass of $850.0 \text{ kg}$. Determine the weight of this beam on Earth where the standard acceleration due to gravity is $g = 9.810 \text{ m/s}^2$. Then, determine its mass on the Moon where the acceleration due to gravity is $g_{m} = 1.620 \text{ m/s}^2$.

An engine block exerts a downward force (weight) of $450.0 \text{ lb}$ on its supports. Determine the mass of the engine block in slugs, and then convert its weight into Newtons (SI). (Use $g = 32.20 \text{ ft/s}^2$ for US Customary units and the conversion factor $1 \text{ lb} = 4.448 \text{ N}$)

A crate with a mass of $50.0 \text{ kg}$ rests on a frictionless horizontal surface. An unbalanced horizontal force $F$ is applied to the crate, causing it to accelerate uniformly at $2.50 \text{ m/s}^2$. Determine the magnitude of the applied force $F$.

A force vector $\mathbf{F}$ with a magnitude of $500.0 \text{ N}$ acts on a bracket at an angle of $\theta = 30.0^\circ$ above the positive x-axis. Determine the horizontal ($F_x$) and vertical ($F_y$) components of this force vector.

Two forces, $\mathbf{P} = 800.0 \text{ N}$ and $\mathbf{Q} = 600.0 \text{ N}$, act on a particle. The angle between the two forces is $120.0^\circ$. Determine the magnitude of the resultant force $\mathbf{R}$ using the Law of Cosines.

Using the same forces from the previous example ($\mathbf{P} = 800.0 \text{ N}$, $\mathbf{Q} = 600.0 \text{ N}$, and $\mathbf{R} = 721.11 \text{ N}$), determine the angle $\alpha$ between the resultant force $\mathbf{R}$ and the force $\mathbf{P}$.

An astronaut with a mass of $80.0 \text{ kg}$ is standing on the surface of the Moon. Calculate the gravitational force of attraction (weight) exerted by the Moon on the astronaut. (Use the mass of the Moon $M_m = 7.342 \times 10^{22} \text{ kg}$, the mean radius of the Moon $R_m = 1737 \text{ km}$, and $G = 66.73 \times 10^{-12} \text{ m}^3/(\text{kg}\cdot\text{s}^2)$)

When studying the orbit of the Earth around the Sun, scientists model both the Earth and the Sun as particles. Why is this particle idealization considered a valid approach in this scenario?

A structural engineer assumes a solid concrete beam is a "rigid body" when determining the support reactions holding it up. In reality, the beam will deform and sag slightly under its own weight. Discuss the justification for using the rigid body assumption.

A car tire resting on a paved road actually contacts the ground over a small rectangular "contact patch" area. However, when solving a free-body diagram of the entire car, engineers model the force from the road as a single "concentrated force" acting at a point. Why is this valid?

A truck is pulling a heavy concrete block using a straight, taut cable attached to the front of the block. If the cable were instead attached to the back of the block, pulling perfectly along the exact same line of action, would the external behavior of the block change?