Thermal Expansion: Solving for Temperature Change

In summary, the question asks for the diameter of a hole in an aluminum plate at different temperatures. Using the formula ΔA = 2αAΔT, where A is the area, α is the coefficient of linear expansion, and ΔT is the change in temperature, we can solve for the diameter of the hole at 199 degrees Celsius by substituting the given values and solving for ΔA. Similarly, we can solve for the temperature at which the diameter of the hole is equal to 1.176 cm by setting ΔA to 0 and solving for ΔT. After converting all measurements to meters, the final answer for part (b) is approximately 1 degree Celsius.
  • #1
izelkay
115
3

Homework Statement


A hole in an aluminum plate has a diameter of 1.178 cm at 23 degrees Celsius. (a) What is the diameter of the hole at 199 degrees Celsius? (b) At what temperature is the diameter of the hole equal to 1.176 cm?

Homework Equations


The coefficient of linear expansion (α) for Aluminum is 24 x 10^-6

Formula to use, I think: ΔA = 2αAΔT
Where A = Area
T = Temperature
α = coefficient of linear expansion


The Attempt at a Solution


I'm having trouble solving part (b).

I keep getting the ΔT to be -502 degrees, which seems like way too much.
What I did first was convert everything from centimeters to meters, and solved for the original area

A = pi(0.0089)²
= 2.49 x 10^-4

And solved for the new Area, A'

A' = pi(0.0088)²
= 2.43 x 10^-4

Subtracted A from A' to get ΔA
A' - A = -6 x 10^-6

Solved for ΔT in formula

ΔA = 2αAΔT

ΔT = ΔA / 2αA

= -6 x 10^-6 / [2(24 x 10^-6)2.49 x 10^-4]
= -502

Again, that seems like too much of a temperature change...can anyone help?
 
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  • #2
Anyone...?
 
  • #3
Never mind, I got it.
 

1. What is thermal expansion?

Thermal expansion is the tendency of matter to expand in volume when heated and contract when cooled. This is due to the increase and decrease in the average distance between atoms or molecules in a substance.

2. Why does thermal expansion occur?

Thermal expansion occurs because of the increase in kinetic energy of particles when heated, causing them to vibrate and take up more space. The opposite happens when the particles lose energy and contract when cooled.

3. What are some real-world examples of thermal expansion?

Some common examples of thermal expansion include the expansion of metals in bridges and buildings due to temperature changes, the expansion of liquids in thermometers, and the expansion of air in hot air balloons.

4. How is thermal expansion measured?

Thermal expansion is typically measured using a coefficient of thermal expansion, which is the change in length or volume of a substance per unit change in temperature. It is usually expressed in units of length per degree Celsius or Kelvin.

5. Can thermal expansion be harmful?

In some cases, thermal expansion can cause damage or deformation to objects that are not designed to withstand significant changes in temperature. For example, it can lead to cracks in roads and buildings, or cause leaks in pipes and containers. However, thermal expansion can also be harnessed for practical use, such as in thermometers and thermostats.

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