A person pushes a lawn mower with a force of 42 pounds. The handle of the lawn mower is 56° above the horizontal. How much downward force is being exerted on the ground? How much horizontal forward force is being exerted? How do these forces change if the handle is closer to horizontal and the same force is applied? If want more of the force on the handle to move the lawn mower forward then I would do what?

To break down the forces acting on the lawn mower, we need to use trigonometry to decompose the applied force into its vertical (downward) and horizontal (forward) components.

Given:
- Force applied (\( F \)) = 42 pounds
- Angle of the handle (\( \theta \)) = 56° above the horizontal

### 1. Downward (vertical) force
The vertical force is given by:
\[
F_{\text{vertical}} = F \cdot \sin(\theta)
\]
Substituting the given values:
\[
F_{\text{vertical}} = 42 \cdot \sin(56^\circ)
\]

Using a calculator, \(\sin(56^\circ) \approx 0.829\):
\[
F_{\text{vertical}} \approx 42 \cdot 0.829 = 34.8 \, \text{pounds}
\]
So, the downward force exerted on the ground is approximately **34.8 pounds**.

### 2. Horizontal (forward) force
The horizontal force is given by:
\[
F_{\text{horizontal}} = F \cdot \cos(\theta)
\]
Substituting the values:
\[
F_{\text{horizontal}} = 42 \cdot \cos(56^\circ)
\]

Using a calculator, \(\cos(56^\circ) \approx 0.559\):
\[
F_{\text{horizontal}} \approx 42 \cdot 0.559 = 23.5 \, \text{pounds}
\]
So, the horizontal forward force is approximately **23.5 pounds**.

### 3. Effect of moving the handle closer to the horizontal
If the handle is positioned closer to the horizontal, the angle \( \theta \) decreases. When this happens:
- The **vertical (downward)** force, \( F_{\text{vertical}} = F \cdot \sin(\theta) \), will decrease because \( \sin(\theta) \) becomes smaller as \( \theta \) decreases.
- The **horizontal (forward)** force, \( F_{\text{horizontal}} = F \cdot \cos(\theta) \), will increase because \( \cos(\theta) \) becomes larger as \( \theta \) decreases.

### 4. How to apply more force forward
To have more of the force move the lawn mower forward, you should **decrease the angle of the handle**, bringing it closer to horizontal. This will increase the horizontal component of the force, as more of the applied force is directed forward rather than downward.

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Let me know if you'd like further details or clarifications.

Here are 5 questions that build on this topic:
1. What is the effect on the horizontal force if the handle angle is increased to 70°?
2. How would the forces change if the applied force were increased to 50 pounds?
3. What angle of the handle would maximize the horizontal force exerted on the lawn mower?
4. How does the force of friction affect the lawn mower’s motion when the horizontal force increases?
5. If the lawn mower is pushed uphill, how do the forces acting on the mower change?

**Tip:** To optimize effort when pushing objects, try to maximize the horizontal force component and minimize the vertical force, since the latter just adds weight to the object without helping its movement.

Learn how to calculate the vertical and horizontal forces exerted on a lawn mower when a 42-pound force is applied at a 56° angle. This solution uses trigonometric functions to decompose the force into components. Explore how changing the angle of the handle impacts the forces. Suitable for Grades 9-12.

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