Force Diagram Problem Solving

Analysis and Solution to Force Diagram Practice Problem #9 by Nathan Morgan.

Question Proposed:

A box slides down a 30 degree ramp with an acceleration of 1.2 meters per second squared. Determine the kinetic friction coefficient between the box and ramp.
Analysis and Solution:

Simplified Diagram of Situation:

The simplified diagram shows the basic concept of what the problem is referring to. A box slides down a ramp inclined at 30 degrees with an acceleration of 1.2 meters per second squared.

Shown above is the available information to be used. The 1.2 meters per second squared acceleration of the box (a sub box, a sub x), angle of ramp's inclination (theta), applied force (F sub app), equality between normal force and horizontal gravity force is explicitly and implicitly stated by the problem. Applied force is assumed to be equal to zero, because the problem doesn't say there is any, the box slides down the ramp only due to the force of gravity. Normal force is concluded to be equal to the box's vertical gravity force component because of the observation that the box, with respect to its self, remains vertically stationary throughout the event (Event being time interval in which the box slides down the ramp). Other preceding information such as acceleration of gravity (g), equation for the box's net x-component force (F sub xnet),  and kinetic force of friction (F sub f) originate from the fundamental laws of forces from physics. It should be noted that the mass (m) of the box is not given. The upcoming diagram will fix any confusion with the significance and labels of discussed force components.

Above is a more detailed diagram of the event including a physical interpretation of the force's direction, along with labels and some relationships between each. The diagram mistakenly labels the normal force to be equal to the product of the force of kinetic friction and force of the kinetic friction coefficient. The variables F sub f and F sub N should be switched. The equation is correctly stated later on in the solving process.

Above is the concluded horizontal net force (F sub xnet) and vertical net force (F sub ynet) of the box. In this situation, horizontal net force is equal to the difference of the horizontal component of the force gravity and the force of kinetic friction between the box and ramp. The vertical net force of the box is assumed to be zero. As previously stated, the box is vertically stationary with respect to itself, meaning that the opposing vertical forces (normal force and the vertical force of gravity) are in balance, therefore equal, meaning the vertical net force is zero.

Above is a relationship between the sin and cos of the angle of inclination and the force of gravity's  components. These were made by using observed characteristics of the force diagram and laws of trigonometry. These equalities were altered for future convenience.

Above is the solving of the problem with the data and equalities discussed. It is noticed that the 

solution was discovered by use of previously theorized equations and algebra. The coefficient of kinetic force of friction was calculated to be approximately 0.44, rounded to the hundredths. The above picture shows how the equations used were concluded.