Question

scenario
angela and carlos are asked to determine the relationship between the normal force on a box of mass m and the angle of incline of the box θ as the box sits at rest on the incline.
using representations
a: the dot at right represents the block on the incline. draw a free - body diagram showing and labeling the forces (not components) exerted on the block. draw the relative lengths of all vectors to reflect the relative magnitudes of all the forces. each force must be represented by a distinct arrow starting on and pointing away from the dot. the dotted line represents the incline.
quantitative analysis
b: start with newtons second law to derive an equation that relates the normal force with the angle of incline. for each line of the derivation, explain in words what you did mathematically. the first line is done for you as an example. express your answer in terms of m, θ, and physical constants as appropriate.
σf_y = ma_y
newtons second law states that the sum of the forces in the \y\ direction will be equal to the mass of the box times the acceleration of the box in the y - direction, therefore:

Explanation:

Step1: Identify forces on block

The forces on the block are the gravitational force $mg$ (down - ward), the normal force $N$ (perpendicular to the incline), and the frictional force (not required for part B but present in real - life to keep the block at rest). For the free - body diagram in part A, draw a dot to represent the block. Draw an arrow straight down labeled $mg$ for the gravitational force, and an arrow perpendicular to the incline labeled $N$ for the normal force. The length of the arrows can be drawn such that the relative magnitudes are represented (in this case, the normal force will be shorter than the gravitational force for non - zero incline angles).

Step2: Apply Newton's second law in y - direction

We choose the y - direction to be perpendicular to the incline. In the y - direction, the acceleration $a_y = 0$ since the block is at rest in the direction perpendicular to the incline. The gravitational force has a component in the y - direction given by $mg\cos\theta$. According to Newton's second law $\sum F_y=ma_y$, and since $a_y = 0$, we have $\sum F_y=N - mg\cos\theta=0$.

Step3: Solve for normal force

We solve the equation $N - mg\cos\theta = 0$ for $N$. By adding $mg\cos\theta$ to both sides of the equation, we get $N = mg\cos\theta$.

Answer:

A: Free - body diagram: Draw a dot for the block. Draw a downward arrow labeled $mg$ and an arrow perpendicular to the incline labeled $N$.
B: $N = mg\cos\theta$