Suppose that a steel hoop could be constructed to fit just around the earth's equator at a temperature of 20.0 Celsius.
What would be the thickness of space between the hoop and the earth if the temperature of the hoop were increased by 0.200 C?
Use 6.38×106m for the radius of the earth, and 1.20×10−5K^-1 for the coefficient of linear expansion of steel.
The increase in hoop length would be
delta L = (2 pi R)*delta T*1.2*10^-5 = 96.2 m
The radius increases by the same factor, 1.2*10^-5*0.2 = 2.4*10^-6.
Multiply that by the original radius.
The new hoop radius is larger by 15.3 m
That is how high the hoop will rise above the surface
To find the thickness of space between the hoop and the earth after the increase in temperature, we need to consider the expansion of the hoop due to the change in temperature.
First, let's find the initial circumference of the hoop at 20.0 Celsius. The circumference of a circle is given by the formula:
C = 2πr
where C is the circumference and r is the radius. Since the hoop fits just around the earth's equator, its circumference at 20.0 Celsius would be the same as the circumference of the earth's equator.
Given the radius of the earth, r = 6.38×10^6m, we can find the initial circumference, C0:
C0 = 2π(6.38×10^6m)
Next, let's consider the expansion of the hoop due to the change in temperature. The change in length of an object can be calculated using the formula:
ΔL = αL0ΔT,
where ΔL is the change in length, α is the coefficient of linear expansion, L0 is the initial length, and ΔT is the change in temperature.
Since we are interested in the thickness, which is the expansion in the radial direction, we can consider the change in radius.
ΔR = αR0ΔT,
where ΔR is the change in radius, α is the coefficient of linear expansion, R0 is the initial radius, and ΔT is the change in temperature.
Given the coefficient of linear expansion of steel, α = 1.20×10^-5K^-1, and the change in temperature, ΔT = 0.200C, we can calculate the change in radius:
ΔR = (1.20×10^-5K^-1)(6.38×10^6m)(0.200C)
Finally, we can find the thickness of space between the hoop and the earth after the temperature increase:
Thickness = C1 - C0
where C1 is the circumference of the hoop at the increased temperature.
To find C1, we need to consider the change in radius, ΔR, and the new radius, R1:
R1 = R0 + ΔR
C1 = 2πR1
Replacing the values and solving:
R1 = 6.38×10^6m + (1.20×10^-5K^-1)(6.38×10^6m)(0.200C)
C1 = 2π(6.38×10^6m + (1.20×10^-5K^-1)(6.38×10^6m)(0.200C))
Finally, we can find the thickness of space:
Thickness = C1 - C0
I hope this helps you understand how to solve the problem!