Write the expression for strain energy. Calculate that for a rod Im long stratched by Imm under a load of 5 kg.

Answer

To solve the problem, we will follow these steps:

### Step 1: Write the expression for strain energy
The expression for strain energy (U) stored in a material when it is deformed is given by:

\[ U = \frac{1}{2} \times \text{stress} \times \text{strain} \times \text{volume} \]

Where:
- Stress = Force / Area
- Strain = Change in length / Original length
- Volume = Area × Length

### Step 2: Substitute the expressions for stress and strain
We can express the strain energy in terms of force (F) and change in length (ΔL):

\[ U = \frac{1}{2} \times \left(\frac{F}{A}\right) \times \left(\frac{\Delta L}{L}\right) \times (A \times L) \]

Here, the area (A) cancels out:

\[ U = \frac{1}{2} \times F \times \Delta L \]

### Step 3: Calculate the force
Given that the load is 5 kg, we can calculate the force (F) using the formula:

\[ F = m \times g \]

Where:
- \( m = 5 \, \text{kg} \)
- \( g = 9.81 \, \text{m/s}^2 \) (acceleration due to gravity)

Calculating the force:

\[ F = 5 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 49.05 \, \text{N} \]

### Step 4: Substitute the values into the strain energy formula
Now we can substitute the values into the strain energy formula. The change in length (ΔL) is given as 1 mm, which we convert to meters:

\[ \Delta L = 1 \, \text{mm} = 1 \times 10^{-3} \, \text{m} \]

Now substituting the values into the strain energy equation:

\[ U = \frac{1}{2} \times 49.05 \, \text{N} \times 1 \times 10^{-3} \, \text{m} \]

### Step 5: Calculate the strain energy
Calculating the strain energy:

\[ U = \frac{1}{2} \times 49.05 \times 1 \times 10^{-3} = \frac{49.05 \times 10^{-3}}{2} = 0.024525 \, \text{J} \]

### Final Answer
The strain energy stored in the rod is approximately:

\[ U \approx 0.0245 \, \text{J} \]