Question

1. Mari pulls a 10 N box behind her at an angle of 70° with the horizontal with a force of 16 N. a. If the surface the box rests on exerts a 1 N frictional force on the box, is Mari pulling hard enough to move the box? Explain. © Success Academy Charter Schools 1
b. Mari then places the block on a 10° ramp (with the same frictional force) and pulls at the same angle with the horizontal. Does that affect the result? Explain.

Answer 1

To determine if Mari is pulling hard enough to move the box, we need to analyze the forces acting on the box.

### Part a:

1. **Identify the Forces Acting on the Box:**
- Pulling Force (F_pull): Mari pulls with a force of 16 N at an angle of 70°.
- Weight of the Box (W): The weight of the box is 10 N acting downward.
- Frictional Force (F_friction): There is a frictional force of 1 N acting in the opposite direction of the pull.

2. **Break Down the Pulling Force:**
The pulling force can be broken into horizontal (F_horizontal) and vertical (F_vertical) components using trigonometry:
- \( F_{horizontal} = F_{pull} \cdot \cos(\theta) = 16 \, N \cdot \cos(70°) \)
- \( F_{vertical} = F_{pull} \cdot \sin(\theta) = 16 \, N \cdot \sin(70°) \)

Calculating these components:
- \( F_{horizontal} \approx 16 \cdot 0.342 = 5.472 \, N \)
- \( F_{vertical} \approx 16 \cdot 0.940 = 15.04 \, N \)

3. **Net Vertical Force:**
The vertical forces acting on the box include the weight of the box and the vertical component of Mari's pulling force. The normal force (N) can be calculated as follows:
- \( N = W - F_{vertical} = 10 \, N - 15.04 \, N \)
Since \( F_{vertical} \) exceeds the weight, the normal force cannot be negative, and this indicates the box would start to lift, but let's consider just the horizontal motion for the moment.

4. **Net Horizontal Force:**
To check if the box moves, we compare the net horizontal force against the friction:
- Net force acting to move the box is \( F_{horizontal} - F_{friction} \)
- Net force = \( 5.472 \, N - 1 \, N = 4.472 \, N \)

Since the net force is positive (4.472 N), Mari is indeed pulling hard enough to overcome friction and initiate movement of the box.

### Part b:

When Mari places the box on a 10° ramp, the situation changes slightly, and we need to analyze the forces again.

1. **Weight Component on the Ramp:**
The weight of the box acting down the ramp is given by:
- \( W_{parallel} = W \cdot \sin(10°) = 10 \, N \cdot \sin(10°) \)
- \( W_{parallel} \approx 10 \cdot 0.174 = 1.74 \, N \)

2. **Normal Force on the Ramp:**
The normal force acting on the box is influenced by the vertical component of the weight and the vertical component of the pulling force:
- The normal force on the ramp can be calculated as:
- \( N = W \cdot \cos(10°) - F_{vertical} \)
- \( N \approx 10 \cdot 0.985 - 15.04 \)
- Since 15.04 N is significantly larger than the weight component acting against the ramp, we can see that the box is still subject to this lifting condition and may not behave under normal conditions as expected.

3. **Net Horizontal Force on the Ramp:**
The net force trying to move the box up the incline now includes gravity pulling it down:
- Net force to move the box = \( F_{horizontal} - (F_{friction} + W_{parallel}) \)
- Net force = \( 5.472 \, N - (1 \, N + 1.74 \, N) = 5.472 \, N - 2.74 \, N = 2.732 \, N \)

### Conclusion:
In both scenarios, Mari is pulling hard enough to move the box, although the dynamics differ slightly due to the ramp's incline and the additional gravitational force acting down the slope. However, Mari's strong pulling force still results in a net positive force, which indicates that she can initiate movement even with the incline taken into account.