Copper telephone wire has essentially no sag between poles 35.0 m apart on a winter
day when the temperature is -20.0℃. What is the length of the wire on a summer day
when the temperature is 35.0℃?
delta L = alpha*L*delta T
is the formula to use. I could not find a consistent value for alpha on the web. I would substitute 35 meters for L, delta T will be 35 - (-20) = 55 and use values in your text/notes for alpha. Post your work if you get stuck.
Well, let me calculate that for you.
Now, we all know that copper expands when it heats up, just like our waistlines during the holiday season. So, as the temperature increases from -20.0℃ to 35.0℃, we can expect the wire to elongate.
To calculate the change in length, we can use the equation:
ΔL = α * L * ΔT
where:
ΔL is the change in length
α is the linear expansion coefficient of copper (approximately 0.000016/℃)
L is the original length of the wire (35.0 m)
ΔT is the change in temperature (35.0℃ - (-20.0℃))
Plugging in the values, we get:
ΔL = 0.000016/℃ * 35.0 m * (35.0℃ - (-20.0℃))
Now let me grab my trusty calculator to do the math.
*beep boop beep*
Okay, after a bit of number crunching, the change in length ΔL is approximately 0.0376 meters.
So, the length of the wire on a summer day at 35.0℃ would be the original length (35.0 m) plus the change in length (0.0376 m), which gives us a total length of approximately 35.0376 meters.
Keep in mind that this calculation assumes a linear expansion, just like my waistline after eating too much cake.
To find the length of the wire on a summer day, we can use the formula for thermal expansion:
ΔL = α * L0 * ΔT
Where:
ΔL is the change in length
α is the coefficient of linear expansion for copper (16.6 x 10^-6 ℃^-1)
L0 is the original length of the wire
ΔT is the change in temperature
Given:
Original length, L0 = 35.0 m
Change in temperature, ΔT = (35.0 - (-20.0)) ℃ = 55.0 ℃
We can substitute these values into the formula:
ΔL = (16.6 x 10^-6 ℃^-1) * (35.0 m) * (55.0 ℃)
Calculating:
ΔL = 16.6 x 10^-6 * 35.0 * 55.0
ΔL = 0.030 m
Now, we can find the length of the wire on a summer day:
Length = L0 + ΔL
Length = 35.0 m + 0.030 m
Length = 35.030 m
Therefore, the length of the wire on a summer day when the temperature is 35.0℃ is approximately 35.030 meters.
To find the length of the wire on a summer day, we need to consider the concept of thermal expansion. When a material is heated, it expands, and when it cools down, it contracts.
The coefficient of linear expansion (α) for copper is approximately 16.5 x 10^-6 ℃^-1. This means that for every degree Celsius increase in temperature, copper wire expands by 16.5 x 10^-6 of its original length.
Given that the temperature changes from -20.0℃ to 35.0℃, we need to calculate the change in temperature (∆T) and apply it to the original length of the wire to determine the new length.
∆T = 35.0℃ - (-20.0℃)
∆T = 55.0℃
Now, we can calculate the change in length (∆L) of the wire:
∆L = α * L * ∆T
Where L is the original length of the wire, which is 35.0m in this case.
Plugging in the values:
∆L = (16.5 x 10^-6 ℃^-1) * (35.0m) * (55.0℃)
∆L ≈ 0.032175m
To find the length of the wire on a summer day, we add the change in length (∆L) to the original length (L).
Length on a summer day = L + ∆L
Length on a summer day = 35.0m + 0.032175m
Length on a summer day ≈ 35.032175m
Therefore, the length of the wire on a summer day, when the temperature is 35.0℃, is approximately 35.032175 meters.