Question

part c: in the spaces below, derive two equations, one in the \y\ direction and one in the \x\ direction, expressing newtons second law using the symbols m, g, a, μ and physical constants as appropriate. for each line of the derivation, explain mathematically what was done (i.e., annotate your derivation). the first line is done for you as an example. 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 car times the acceleration of the car in the \y\ direction, therefore:

Explanation:

Step1: Analyze forces in y - direction

In the y - direction, the normal force $N$ and the component of gravitational force act. Assuming the surface is horizontal, the gravitational force is $mg$ acting downwards and the normal force $N$ acting upwards. So, $\sum F_y=N - mg$. According to Newton's second law $\sum F_y = ma_y$, we have $N - mg=ma_y$.

Step2: Analyze forces in x - direction

Let the frictional force be $f=\mu N$ and an external force $F$ act in the x - direction. The net force in the x - direction is $\sum F_x=F - f$. Since $f = \mu N$ and from the y - direction $N=mg + ma_y$, then $\sum F_x=F-\mu(mg + ma_y)$. According to Newton's second law $\sum F_x=ma_x$, so $F-\mu(mg + ma_y)=ma_x$.

Answer:

In the y - direction: $N - mg=ma_y$
In the x - direction: $F-\mu(mg + ma_y)=ma_x$