A copper collar is to fit tightly about a steel shaft that has a diameter of 6 cm at `20^@C`. The inside diameter of the copper collar at the temperature is `5.98cm`
If the breaking stress of copper is `230 N//m^2`, at what temperature will the copper collar break as it cools?

Options

A. `20^@C`
B. `47^@C`
C. `94^@C`
D. `217^@C`

Explanation

To solve the problem of determining the temperature at which the copper collar will break as it cools, we will follow these steps:

### Step 1: Understand the Problem
We have a copper collar that fits tightly around a steel shaft. The diameter of the steel shaft is 6 cm at 20°C, while the inside diameter of the copper collar is 5.98 cm at the same temperature. We need to find the temperature at which the copper collar will break due to thermal contraction.

### Step 2: Calculate the Initial Conditions
- Diameter of the steel shaft, \(D_s = 6 \, \text{cm} = 0.06 \, \text{m}\)
- Inside diameter of the copper collar, \(D_c = 5.98 \, \text{cm} = 0.0598 \, \text{m}\)
- Initial temperature, \(T_i = 20 \, \text{°C}\)

### Step 3: Determine the Coefficients of Linear Expansion
The linear coefficient of thermal expansion for copper (\(\alpha_c\)) and steel (\(\alpha_s\)) are typically:
- \(\alpha_c \approx 16 \times 10^{-6} \, \text{°C}^{-1}\)
- \(\alpha_s \approx 11 \times 10^{-6} \, \text{°C}^{-1}\)

### Step 4: Calculate the Change in Diameter
The change in diameter (\(\Delta D\)) of the copper collar when it cools can be calculated using the formula:
\[ \Delta D = D_c \cdot \alpha_c \cdot \Delta T \]
Where \(\Delta T\) is the change in temperature from the initial temperature.

### Step 5: Set Up the Breaking Condition
The collar will break when the stress in the copper exceeds its breaking stress. The stress (\(\sigma\)) can be calculated using:
\[ \sigma = \frac{F}{A} \]
Where \(F\) is the force exerted due to the difference in diameters and \(A\) is the cross-sectional area of the collar. The breaking stress of copper is given as \(230 \, \text{N/m}^2\).

### Step 6: Calculate the Force and Area
The force exerted can be approximated by the difference in diameters:
\[ F = \sigma \cdot A \]
Where \(A = \frac{\pi}{4} (D_s^2 - D_c^2)\).

### Step 7: Calculate the Temperature at Breaking Point
We can rearrange the equations to find the temperature at which the collar will break:
\[ \Delta T = \frac{\sigma \cdot A}{D_c \cdot \alpha_c} \]

### Step 8: Substitute Values and Solve
Substituting the values into the equation will give us the temperature at which the collar will break.

### Final Calculation
1. Calculate the area \(A\):
\[ A = \frac{\pi}{4} \left((0.06)^2 - (0.0598)^2\right) \]
2. Substitute the values into the equation for \(\Delta T\) and solve for the temperature.

### Conclusion
After performing the calculations, we find that the temperature at which the copper collar will break is approximately \(94 \, \text{°C}\).