Question

scenario: carlos pushes a block of mass, m, across a rough horizontal surface at a constant speed by applying a force, f, directly to the right. using representations part a: the dot at right represents the block. draw a free - body diagram showing and labeling all 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. quantitative analysis part b: blake is asked to use newtons second law to derive an equation that relates the force of gravity and the normal force from the surface exerted on the block. annotate his derivation by filling in the right - hand side of the table below. for each line of the derivation, explain in words what was done mathematically. the first line is done for you. $sum f_y=ma_y$ we start with newtons second law, which says that the sum of all the forces exerted on an object is equal to the objects mass times the objects acceleration. $f_n - f_g=ma_y$ $f_n - f_{mg}=m\times0$ $f_n - f_{mg}=0$ $f_n = f_{mg}$

Explanation:

Step1: Identify forces in y - direction

In the vertical (y -) direction, the normal force $F_N$ acts upward and the force of gravity $F_{mg}=mg$ acts downward. According to Newton's second law $\sum F_y = ma_y$.

Step2: Analyze acceleration in y - direction

Since the block is moving horizontally at a constant speed, there is no acceleration in the vertical direction, so $a_y = 0$. Substituting into $\sum F_y = ma_y$, we get $F_N - F_{mg}=ma_y$.

Step3: Substitute acceleration value

As $a_y = 0$, the equation $F_N - F_{mg}=ma_y$ becomes $F_N - F_{mg}=m\times0 = 0$.

Step4: Solve for the relationship

Adding $F_{mg}$ to both sides of the equation $F_N - F_{mg}=0$, we find that $F_N=F_{mg}$.

Answer:

The normal force $F_N$ exerted by the surface on the block is equal to the force of gravity $F_{mg}$ acting on the block, i.e., $F_N = F_{mg}$ where $F_{mg}=mg$ (with $m$ being the mass of the block and $g$ the acceleration due to gravity).

For the free - body diagram part (PART A):

1. Draw a dot to represent the block.
2. Draw a vertical arrow pointing upward to represent the normal force $F_N$. Label it $F_N$.
3. Draw a vertical arrow of equal length pointing downward to represent the force of gravity $F_{mg}$. Label it $F_{mg}$.
4. Draw a horizontal arrow to the right to represent the applied force $F$. Label it $F$.
5. Draw a horizontal arrow to the left of shorter length (assuming kinetic friction) to represent the frictional force $F_f$. Label it $F_f$. The lengths of the arrows should reflect the relative magnitudes of the forces. Since the block is moving at a constant speed, $F = F_f$ and $F_N=F_{mg}$.