Question

A surveyor uses a steel measuring tape that is exactly 50.000 m long at a temperature of 20°C. What is the length on a hot summer day when the temperature is 35°C? The average coefficient of linear expansion (α) for steel is 1.2 x 10^-5 K^-1.

Answer 1

On a hot summer day with a temperature of
`\(35^\circ \text{C}\)`, the steel measuring tape, initially 50.000 m long at 20°C, expands by 0.009 m due to its coefficient of linear expansion
`(\(\alpha = 1.2 * 10^(-5) \, \text{K}^(-1)\))`. The final length of the tape on the hot day is approximately 50.009 m.

Step-by-step explanation:

To calculate the change in length
`(\(\Delta L\))` of the steel measuring tape due to a change in temperature, you can use the formula:

`\[ \Delta L = L_0 \cdot \alpha \cdot \Delta T \]`

where:

-
`\( \Delta L \)` is the change in length,

-
`\( L_0 \)` is the original length of the tape,

-
`\( \alpha \)` is the coefficient of linear expansion,

-
`\( \Delta T \)` is the change in temperature.

Given:

-
`\( L_0 = 50.000 \, \text{m} \)` (original length),

-
`\( \alpha = 1.2 * 10^(-5) \, \text{K}^(-1) \)` (coefficient of linear expansion),

-
`\( \Delta T = T_{\text{final}} - T_{\text{initial}} \)` (change in temperature).

Substitute the values and solve for \( \Delta L \):

`\[ \Delta L = (50.000 \, \text{m}) \cdot (1.2 * 10^(-5) \, \text{K}^(-1)) \cdot (35^\circ \text{C} - 20^\circ \text{C}) \]`

`\[ \Delta L = (50.000 \, \text{m}) \cdot (1.2 * 10^(-5) \, \text{K}^(-1)) \cdot (15^\circ \text{C}) \]`

`\[ \Delta L = 0.009 \, \text{m} \]`

Now, to find the length on the hot summer day
`(\(L_{\text{final}}\))`, you add the change in length to the original length:

`\[ L_{\text{final}} = L_0 + \Delta L \]`

`\[ L_{\text{final}} = 50.000 \, \text{m} + 0.009 \, \text{m} \]`

`\[ L_{\text{final}} = 50.009 \, \text{m} \]`

So, on a hot summer day when the temperature is
`\(35^\circ \text{C}\)`, the length of the steel measuring tape would be approximately
`\(50.009 \, \text{m}\)`.

Answer 2

Using the formula for linear thermal expansion, the steel measuring tape that is precisely 50.000 meters at 20°C would extend by 9 mm to a length of 50.009 meters at 35°C.

Step-by-step explanation:

The subject of this question involves the concept of linear thermal expansion in Physics, where materials expand in response to temperature changes. To calculate the change in length of the steel measuring tape from 20°C to 35°C, we use the formula ΔL = αLΔT, where ΔL is the change in length, α is the coefficient of thermal expansion, L is the original length, and ΔT is the change in temperature.

Given the coefficient of linear expansion for steel as 1.2 x 10-5 K-1 and the original length as 50.000 m:

ΔT = 35°C - 20°C = 15°C

ΔL = (1.2 x 10-5 K-1)(50.000 m)(15°C)

ΔL = (1.2 x 10-5)(50.000)(15)

ΔL = 9 x 10-3 m or 9 mm

The steel tape would be 50.009 m long on a hot summer day when the temperature is 35°C.

The complete question is: A surveyor uses a steel measuring tape that is exactly 50.000 m long at a temperature of 20°C. What is the length on a hot summer day when the temperature is 35°C? The average coefficient of linear expansion (α) for steel is 1.2 x 10^-5 K^-1. is: