A force \(\vec{F}\) applied at the point \(x=2.0 \mathrm{m}, y=0 \mathrm{m}\) produces a torque \(4.6 \hat{k}\) N.m about the origin. If the \(x\) -component of \(\vec{F}\) is 3.1 \(\mathrm{N}\), what angle does it make with the \(x\) -axis?

Understanding force vectors is integral to grasping many physics concepts. A force vector, denoted as \(\vec{F}\), is a representation of both the magnitude and the direction of a force applied to an object. It has two components: one in the x-direction (horizontal), \(F_x\), and one in the y-direction (vertical), \(F_y\). These components help us understand how the force influences the motion of the object in two dimensions. When a force is applied at an angle, not directly along the axes, we can use trigonometry to resolve the vector into its x and y components.

In our exercise, we see an application of this concept, where the force applied has a component along the x-axis, known as \(F_x\), and we are tasked with finding the y-component, \(F_y\), to determine the angle of the force vector with respect to the x-axis. The relationship between the two components and the angle \(\theta\) can be described using basic trigonometric functions, which is key to solving this type of problem.

Angular momentum is a measure of the quantity of rotation an object possesses. Similar to linear momentum but for rotational movement, angular momentum (\(\vec{L}\)) is dependent on the object's moment of inertia and angular velocity. One of the cornerstone principles of angular momentum is its conservation — in an isolated system, the total angular momentum remains constant if there is no external torque.

This principle is not directly part of our exercise, but understanding it puts into context why it's crucial to know the torque when discussing rotational dynamics. Torque is, effectively, the rotational equivalent of a force that causes changes in an object's angular momentum.

Torque, represented by \(\tau\), is a measure of the force that can cause an object to rotate about an axis. The calculation of torque involves both the magnitude of the force and its position relative to the rotation point, or, more precisely, the length of the lever arm \(\vec{r}\) and the angle \(\theta\) at which the force is applied. The mathematical expression for torque is \(\tau = \vec{r} \times \vec{F}\) or \(\tau = |r||F| \sin{\theta} \hat{k}\), where \(\times\) symbolizes the cross product between the position vector and the force vector, and \(\hat{k}\) indicates the direction of the torque (out of the page or into the page for a three-dimensional perspective).

In our exercise scenario, we utilize the formula to calculate the y-component of the force, given the torque and the known x-component of the force. This is a practical application of torque calculation, providing a pathway to discovering unknown variables within a physical situation.