A vertical steel beam at the base of buiolding is 6.0 m tall, thas a mass of 300 kg, and supports a load of `3.0xx10^(5)N`. If the beam's temperature decreases by`4.0^(@)C`, calculate the change in its internal energy using the facts that for steel `c_(p)` is `0.11 "kcal"//kg^(@)` and the coefficient of the linear expansion is `11xx10^(-6).^(@)C^(-1)`

Answer

To solve the problem of finding the change in internal energy of a vertical steel beam when its temperature decreases, we can follow these steps:

### Step 1: Calculate the change in length due to temperature change

The formula for the change in length (ΔL) due to temperature change is given by:

\[ \Delta L = L \cdot \alpha \cdot \Delta T \]

Where:
- \(L\) = original length of the beam = 6.0 m
- \(\alpha\) = coefficient of linear expansion for steel = \(11 \times 10^{-6} \, ^\circ C^{-1}\)
- \(\Delta T\) = change in temperature = -4.0 °C (since it is a decrease)

Substituting the values:

\[ \Delta L = 6.0 \, \text{m} \cdot (11 \times 10^{-6} \, ^\circ C^{-1}) \cdot (-4.0 \, ^\circ C) \]

Calculating this gives:

\[ \Delta L = 6.0 \cdot 11 \cdot (-4) \times 10^{-6} = -264 \times 10^{-6} \, \text{m} = -0.000264 \, \text{m} \]

### Step 2: Calculate the change in internal energy due to temperature change

The change in internal energy (ΔU) due to temperature change can be calculated using the formula:

\[ \Delta U = m \cdot c_p \cdot \Delta T \]

Where:
- \(m\) = mass of the beam = 300 kg
- \(c_p\) = specific heat capacity of steel = \(0.11 \, \text{kcal/kg} \, ^\circ C\)
- \(\Delta T\) = change in temperature = -4.0 °C

Converting \(c_p\) to Joules (1 kcal = 4184 J):

\[ c_p = 0.11 \, \text{kcal/kg} \, ^\circ C \times 4184 \, \text{J/kcal} = 460.24 \, \text{J/kg} \, ^\circ C \]

Now substituting the values:

\[ \Delta U = 300 \, \text{kg} \cdot 460.24 \, \text{J/kg} \, ^\circ C \cdot (-4.0 \, ^\circ C) \]

Calculating this gives:

\[ \Delta U = 300 \cdot 460.24 \cdot (-4) = -553,488 \, \text{J} \]

### Step 3: Convert the change in internal energy to kilocalories

To convert Joules to kilocalories:

\[ \Delta U = -553,488 \, \text{J} \div 4184 \, \text{J/kcal} \approx -132.2 \, \text{kcal} \]

### Final Answer

The change in internal energy of the steel beam when the temperature decreases by 4.0 °C is approximately **-132.2 kcal**.

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