In a lush, vibrant field where crops sway gently in the breeze, a bushy plant stands * proudly. Its stem, slender but resilient, measures precisely 6 mm in diameter. The task at hand is to sever this stem using a powerful impact force. To add a touch of precision to the process, the force is to be applied at a specific height-50 mm above the soil surface, where the stem is particularly vulnerable.

The plant's stem is more than just a simple structure; it's an intricate combination of strength and flexibility. The modulus of elasticity of this stem, which describes its ability to resist deformation under stress, is 1500 N/m². Additionally, the ultimate tensile strength, which indicates the maximum stress the stem can withstand before failing, is 35 N/m².

As the moment of truth approaches, the challenge is to calculate the precise force required to cause the stem to fail due to bending. With it approximated as 3.14, this calculation will determine how much force is needed to overcome the stem's natural resilience and achieve the desired result. This calculation is not just a matter of numbers, but a crucial step in understanding the delicate balance of nature and the mechanics involved in harvesting this bushy crop.

To calculate the precise force required to cause the plant's stem to fail due to bending, we'll employ the bending stress formula from mechanics of materials. The calculation involves determining the bending moment and then the required force that will induce a stress equal to the ultimate tensile strength at the point where the stem is severed.

The following steps outline the solution:

### **1. Understand the Given Parameters:**
- Diameter of the stem \( d = 6 \, \text{mm} = 0.006 \, \text{m} \)
- Height where the force is applied \( h = 50 \, \text{mm} = 0.05 \, \text{m} \)
- Modulus of elasticity \( E = 1500 \, \text{N/m}^2 \) (not directly needed here)
- Ultimate tensile strength \( \sigma_u = 35 \, \text{N/m}^2 \)
- Approximated \(\pi = 3.14\)

### **2. Bending Stress and Force Calculation:**

The maximum bending stress \( \sigma \) experienced by a beam (or in this case, the plant stem) subjected to a bending moment \( M \) is calculated using:

\[
\sigma = \frac{M \cdot c}{I}
\]

where:
- \( M \) is the bending moment
- \( c \) is the distance from the neutral axis to the outer fiber (half of the diameter, \( \frac{d}{2} \))
- \( I \) is the second moment of area for a circular cross-section, given by \( I = \frac{\pi d^4}{64} \)

#### **Step 1: Calculate \( I \) (Moment of Inertia)**
\[
I = \frac{\pi d^4}{64}
\]
Substituting \( d = 0.006 \, \text{m} \):
\[
I = \frac{3.14 \times (0.006)^4}{64}
\]

#### **Step 2: Calculate \( c \)**
\[
c = \frac{d}{2} = \frac{0.006}{2} = 0.003 \, \text{m}
\]

#### **Step 3: Determine the bending moment \( M \)**
The bending moment \( M \) at the base where the force \( F \) is applied:
\[
M = F \times h
\]

To find the force \( F \), we set the bending stress \( \sigma \) equal to the ultimate tensile strength \( \sigma_u \):
\[
\sigma_u = \frac{M \cdot c}{I}
\]
Substituting \( M = F \times h \):
\[
\sigma_u = \frac{F \cdot h \cdot c}{I}
\]
Solving for \( F \):
\[
F = \frac{\sigma_u \cdot I}{h \cdot c}
\]

Let’s perform these calculations.The calculated force required to cause the stem to fail due to bending is approximately \(1.48 \times 10^{-5} \, \text{N}\). This is the precise force needed to overcome the stem's natural resilience and achieve the desired result of severing it at the given height.

Would you like me to provide more details or answer any questions about this calculation?

### Related Questions:
1. How does changing the height at which force is applied affect the required force for breaking the stem?
2. What role does the modulus of elasticity play in determining the stem's bending behavior?
3. How would the force requirement change if the stem's diameter increased?
4. What are some real-world applications where understanding bending stress is crucial?
5. How can this approach be adapted for non-cylindrical stems or structures?

### Tip:
When dealing with bending problems, always remember that the distance from the point of force application (lever arm) significantly impacts the bending moment and thus the required force.

In a lush, vibrant field where crops sway gently in the breeze, a bushy plant stands * proudly. Its stem, slender but resilient, measures precisely 6 mm in diameter. The task at hand is to sever this stem using a powerful impact force. To add a touch of precision to the process, the force is to be applied at a specific height-50 mm above the soil surface, where the stem is particularly vulnerable. The plant's stem is more than just a simple structure; it's an intricate combination of strength and flexibility. The modulus of elasticity of this stem, which describes its ability to resist deformation under stress, is 1500 N/m². Additionally, the ultimate tensile strength, which indicates the maximum stress the stem can withstand before failing, is 35 N/m². The challenge is to calculate the precise force required to cause the stem to fail due to bending.

Discover how to calculate the force required to sever a plant stem with a 6 mm diameter at 50 mm height, considering the modulus of elasticity of 1500 N/m² and ultimate tensile strength of 35 N/m². This problem uses the bending stress formula and is ideal for undergraduate students in engineering or physics.

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