How Does Temperature Affect the Expansion of a Rectangular Plate?

[Rblswimmer456]

Homework Statement

A rectangular plate has an area of 76 cm2.
If the temperature increases by 199 ◦C,
calculate the increase in the area. Use
2 × 10−6 (◦C)−1 as an average coefficient of
linear expansion. Ignore terms of second or-
der.
Answer in units of cm2.

Homework Equations

I looked through my physics book and there wasn't really anything about temperature vs area and I looked online but I didn't find much so I thought I could just you use the equation:

delta(L)= (alpha)(initial L)delta(T) and I just replaced L with A..so it read...
delta(A)=(alpha)(initial A)delta(T)...and alpha being the coefficent of linear expansion

The Attempt at a Solution

delta(A)=(2e-6)(76cm²)(199)

..and this was incorrect...any help would be appreciated to point me in the right direction...thank you!

 

[tiny-tim]

Welcome to PF!

Hi Rblswimmer456! Welcome to PF!

(ahve a delta: ∆ and a degree: ° and try using the X² tag just above the Reply box )

… Use
2 × 10−6 (◦C)−1 as an average coefficient of linear expansion.
…
delta(A)=(2e-6)(76cm²)(199)

"linear expansion" means that every length is multiplied by 2 10^-6 per °C …

so an area is multiplied by … ?

 

[Rblswimmer456]

so since its a "rectangular plate" would 2x10^-6 be multiplied by 4 to get the total linear expansion?

 

[tiny-tim]

oops!

oops! just realized my last post was wrong …

(not completely awake yet! :zzz:)

should have been …

"linear expansion" means that every length is increased by 2 10-6 per °C …

so every length is multiplied by 1.000002 per °C

so an area is multiplied by … ?

so since its a "rectangular plate" would 2x10^-6 be multiplied by 4 to get the total linear expansion?

(did you mean "linear" ? )

no, nor the area expansion

if the length was multiplied by 2, then the area would be multiplied by 4

if the length is multiplied by 1.000002, then the area is multiplied by … ?

 

[rwisz]

Your attempt at using the equation for linear expansion is correct. However, there are a few things to consider. First, make sure to convert the temperature from degrees Celsius to Kelvin (since the coefficient of linear expansion is in K^-1). So the temperature change would be 199 K.

Next, the coefficient of linear expansion is typically given in units of K^-1, not °C^-1. So you would need to convert the given value of 2 × 10^-6 (°C)^-1 to 2 × 10^-6 K^-1.

Finally, the equation you used gives the change in length, not area. To find the change in area, you can use the equation A = L x W, where L is the original length and W is the original width. So the equation would be:

delta(A) = (alpha)(initial A)(delta(T))

= (2 × 10^-6 K^-1)(76 cm^2)(199 K)

= 3.03 cm^2

Therefore, the increase in area would be 3.03 cm^2.