A crate of mass $100\ \mathrm{kg}$ is at rest on a horizontal floor. The coefficient of static friction between the crate and the floor is $0.4$, and the coefficient of kinetic friction is $0.3$. A force $\textbf F$ of magnitude $344\ \mathrm{N}$ is then applied to the crate, parallel to the floor. Which of the following is true?
A) The crate will accelerate across the floor at $0.5\ \mathrm{\displaystyle \frac{m}{\ s^2}}$.
B) The static friction force, which is the reaction force to $\textbf F$ as guaranteed by Newton’s Third Law, will also have a magnitude of $344\ \mathrm{N}$.
C) The crate will slide across the floor at a constant speed of $0.5\ \mathrm{\displaystyle \frac{m}{s}}.$
D) The crate will not move.
A) The crate will accelerate across the floor at $0.5\ \mathrm{\displaystyle \frac{m}{\ s^2}}$.
B) The static friction force, which is the reaction force to $\textbf F$ as guaranteed by Newton’s Third Law, will also have a magnitude of $344\ \mathrm{N}$.
C) The crate will slide across the floor at a constant speed of $0.5\ \mathrm{\displaystyle \frac{m}{s}}.$
D) The crate will not move.
The maximum force which static friction can exert on the crate is
$$\mu_{\mathrm{s}}F_{\mathrm{N}}=\mu_{\mathrm{s}}F_{\mathrm{w}}$$
$$\mu_{\mathrm{s}}F_{\mathrm{N}}=\mu_{\mathrm{s}}mg$$
$$\mu_{\mathrm{s}}F_{\mathrm{N}}=(0.4)(100\ \mathrm{kg})\left(10\ \mathrm{\displaystyle \frac{N}{kg}}\right)$$
$$\mu_{\mathrm{s}}F_{\mathrm{N}}=400\ \mathrm{N}$$
Since the force applied to the crate is only $344\ \mathrm{N}$, static friction is able to apply that same magnitude of force on the crate, keeping it stationary. Choice (B) is incorrect because the static friction force is not the reaction force to $\textbf F$; both $\textbf F$ and $\textbf F_{\mathrm{f(static)}}$ act on the same object (the crate) and therefore cannot form an action/reaction pair.