Let's break this problem down and solve it step by step:

Given dataLet's break this problem down and solve it step by step:

Given

• Block $A$ weighs $1.30 \, \text{N}$.
• Block $B$ weighs $3.60 \, \text{N}$.
• Coefficient of kinetic friction between all surfaces: $\mu_k = 0.310$.
• Block $A$ is held at rest, and we need to find the horizontal force $\vec{F}$ necessary to drag block $B$ to the left at a constant speed.

Step 1: Calculate normal forces

The normal force exerted on each block is equal to the weight of the block because the forces are vertical.

• For block $A$, the normal force is $N_A = 1.30 \, \text{N}$.
• For block $B$, the normal force is the sum of its own weight and the weight of block $A$, since block $A$ is sitting on top of block $B$. $N_B = 3.60 \, \text{N} + 1.30 \, \text{N} = 4.90 \, \text{N}$

Step 2: Calculate friction forces

The kinetic friction force is given by: $f_k = \mu_k \cdot N$

• Friction force between block $A$ and block $B$: $f_{A/B} = \mu_k \cdot N_A = 0.310 \cdot 1.30 \, \text{N} = 0.403 \, \text{N}$

• Friction force between block $B$ and the surface: $f_{B/\text{surface}} = \mu_k \cdot N_B = 0.310 \cdot 4.90 \, \text{N} = 1.519 \, \text{N}$

Step 3: Analyze forces acting on block $B$

Since block $A$ is held at rest, the force required to drag block $B$ at a constant speed is the sum of the friction forces opposing the motion:

$F_{\text{net}} = f_{A/B} + f_{B/\text{surface}} = 0.403 \, \text{N} + 1.519 \, \text{N} = 1.922 \, \text{N}$

Final Answer:

The magnitude of the horizontal force $\vec{F}$ necessary to drag block $B$ to the left at constant speed is $1.922 \, \text{N}$.

Would you like a deeper explanation or further details? Here are some related questions:

1. How would the force change if block $A$ were allowed to slide freely?
2. What role does the coefficient of friction play in determining the force?
3. How would increasing the weight of block $B$ affect the required force?
4. How do we calculate normal forces in more complex systems like inclined planes?
5. What happens to the frictional force if the surface between $B$ and the ground changes?

Tip: When objects are in contact, the normal force and friction play crucial roles in determining how much force is needed to move them at a constant velocity. Always account for all points of contact.