Question

below are eight crates of different mass. (figure 1) the crates are attached to massless ropes, as indicated in the picture, where the ropes are marked by letters. each crate is being pulled to the right at the same constant speed. the coefficient of kinetic friction between each crate and the surface on which it slides is the same for all eight crates. part a rank the ropes on the basis of the force each exerts on the crate immediately to its left. rank from largest to smallest. to rank items as equivalent, overlap them.

Explanation:

Step1: Analyze forces on each crate

Since the crates are moving at a constant speed, the net - force on each crate is zero. For a single crate of mass $m$, the frictional force $f=\mu_k N=\mu_kmg$ (where $\mu_k$ is the coefficient of kinetic friction, $N$ is the normal force, and on a horizontal surface $N = mg$). The force exerted by the rope on the crate to the left must balance the frictional force.

Step2: Calculate frictional force for each case

For crate A with $m_A=(3 + 2+1)\text{ kg}=6\text{ kg}$, $F_A=\mu_k m_Ag$. For crate B with $m_B=(2 + 1)\text{ kg}=3\text{ kg}$, $F_B=\mu_k m_Bg$. For crate C with $m_C = 1\text{ kg}$, $F_C=\mu_k m_Cg$. For crate D with $m_D = 1\text{ kg}$, $F_D=\mu_k m_Dg$. For crate E with $m_E=(1 + 2)\text{ kg}=3\text{ kg}$, $F_E=\mu_k m_Eg$. For crate F with $m_F=(2 + 3)\text{ kg}=5\text{ kg}$, $F_F=\mu_k m_Fg$. For crate G with $m_G = 3\text{ kg}$, $F_G=\mu_k m_Gg$. For crate H with $m_H = 5\text{ kg}$, $F_H=\mu_k m_Hg$.

Step3: Rank the forces

The forces are proportional to the mass of the crates to the right of the rope. Ranking the masses of the crates to the right of each rope from largest to smallest:

• For rope A: mass to the right is $6\text{ kg}$.
• For rope B: mass to the right is $3\text{ kg}$.
• For rope C: mass to the right is $1\text{ kg}$.
• For rope D: mass to the right is $1\text{ kg}$.
• For rope E: mass to the right is $3\text{ kg}$.
• For rope F: mass to the right is $5\text{ kg}$.
• For rope G: mass to the right is $3\text{ kg}$.
• For rope H: mass to the right is $5\text{ kg}$.

The ranking from largest to smallest force exerted by the ropes is: $A>F = H>B = E = G>C = D$.

Answer:

Largest: A
Second - largest: F, H (overlapped)
Third - largest: B, E, G (overlapped)
Smallest: C, D (overlapped)