In Step 6 "Analysis", for Case 4, how does the Friction Force compare to the quantity of Mass times Acceleration? The Friction Force is equal to the quantity of Mass times Acceleration The Friction Force and the quantity of Mass times Acceleration are both equal to zero The Friction Force is less than the quantity of Mass times Acceleration The Friction Force is greater than the quantity of Mass times Acceleration

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To determine how the friction force compares to the product of mass and acceleration ($$ F_{\text{friction}} $$ vs. $$ m \times a $$) in **Step 6: Analysis for Case 4**, let's break down the possible scenarios based on Newton's Second Law of Motion:

$$
\sum F = m \times a
$$

Here, $$ \sum F $$ represents the net force acting on an object, which can include various forces such as applied force, friction, tension, etc.

### Understanding the Options:

1. **The Friction Force is equal to the quantity of Mass times Acceleration ($$ F_{\text{friction}} = m \times a $$):**
   - **Possible Scenario:** This occurs when friction is the only horizontal force acting on the object. In this case, friction provides the necessary force to accelerate the mass. Thus, $$ F_{\text{friction}} = m \times a $$.

2. **The Friction Force and the quantity of Mass times Acceleration are both equal to zero ($$ F_{\text{friction}} = 0 $$ and $$ m \times a = 0 $$):**
   - **Possible Scenario:** This happens when there is no relative motion or impending motion between surfaces (static equilibrium). If there's no acceleration, then $$ a = 0 $$, which implies $$ F_{\text{friction}} = 0 $$ if no other horizontal forces are acting.

3. **The Friction Force is less than the quantity of Mass times Acceleration ($$ F_{\text{friction}} < m \times a $$):**
   - **Possible Scenario:** This could occur if there are additional forces acting in the direction of acceleration besides friction. For example, an applied force greater than friction would result in $$ F_{\text{friction}} < m \times a $$.

4. **The Friction Force is greater than the quantity of Mass times Acceleration ($$ F_{\text{friction}} > m \times a $$):**
   - **Possible Scenario:** This might occur if friction opposes a force sufficiently strong to cause acceleration, potentially leading to deceleration or preventing motion.

### Conclusion for **Case 4**:

Assuming **Case 4** refers to a scenario where friction is the primary force causing acceleration (and no other horizontal forces are acting), the friction force would need to provide the exact amount of force required to accelerate the mass. This aligns with the first option.

**Therefore, the correct comparison is:**

**The Friction Force is equal to the quantity of Mass times Acceleration.**

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**Answer:**  
The Friction Force is equal to the quantity of Mass times Acceleration
The Friction Force is equal to the quantity of Mass times Acceleration

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