Let's simplify the process step by step for filling in the blanks using the given diagram and friction equation:

Key Formulas:

1. Friction force:
   $F_{\text{frict}} = \mu F_{\text{norm}}$

2. Net force (Newton's Second Law):
   $F_{\text{net}} = m \cdot a = F_{\text{app}} - F_{\text{frict}}$

Steps to Solve Each Case:

Case 1: $a = 0 \, \text{m/s}^2$

• If acceleration $a = 0$, the net force is zero: $F_{\text{net}} = 0 \implies F_{\text{app}} = F_{\text{frict}}$
• Substitute $F_{\text{frict}} = \mu F_{\text{norm}}$: $\mu = \frac{F_{\text{frict}}}{F_{\text{norm}}}$
• Use the values:
  $F_{\text{app}} = 10 \, \text{N}, F_{\text{norm}} = 12 \, \text{N}$: $F_{\text{frict}} = F_{\text{app}} = 10 \, \text{N}$ $\mu = \frac{10}{12} = 0.833$

Case 2: $a = 5 \, \text{m/s}^2 \, (\text{left})$

• Acceleration is $a = 5 \, \text{m/s}^2$ and directed left.

• Use $F_{\text{net}} = m \cdot a$:
  Let the net force be $F_{\text{net}} = F_{\text{frict}} - F_{\text{app}}$ (since the block is accelerating left).

• Rearrange for $F_{\text{frict}}$: $F_{\text{frict}} = F_{\text{app}} + F_{\text{net}}$

• To find $F_{\text{net}}$, calculate: F_{\text{net}} = m \cdot a = 12 \cdot 5 = 60 \, \text{N} \, (\text{mass is 12 N assuming g = 1 m/s^2}) $F_{\text{frict}} = 16 + 60 = 76 \, \text{N}$

Case 3: $\mu = 0.500$

• Friction force is: $F_{\text{frict}} = \mu F_{\text{norm}} = 0.500 \cdot 12 = 6 \, \text{N}$
• Net force: $F_{\text{net}} = F_{\text{app}} - F_{\text{frict}} = 30 - 6 = 24 \, \text{N}$
• Acceleration: $a = \frac{F_{\text{net}}}{m} = \frac{24}{12} = 2 \, \text{m/s}^2$

Final Answers:

1. Case 1:

   • $F_{\text{frict}} = 10 \, \text{N}$
   • $\mu = 0.833$

2. Case 2:

   • $F_{\text{frict}} = 76 \, \text{N}$
   • $a = 5 \, \text{m/s}^2 \, (\text{left})$

3. Case 3:

   • $F_{\text{frict}} = 6 \, \text{N}$
   • $a = 2 \, \text{m/s}^2$

Let me know if you want more detailed steps for any part!

Related Questions:

1. How does the coefficient of friction $\mu$ affect the motion of an object?
2. What happens when $F_{\text{app}} < F_{\text{frict}}$?
3. Why is the normal force $F_{\text{norm}}$ important in friction calculations?
4. How can we experimentally measure $\mu$ in a real-world scenario?
5. Why is the direction of acceleration opposite in Case 2?

Tip:

Always start by calculating the net force $F_{\text{net}}$, then solve for $F_{\text{frict}}$ and $\mu$.