An alumium rod when measured with a steel scale both being at 25 degree celcius appears to be 1 meter long. If the scale is correct at 0 degree celcius, what is the true length of the rod at 25 degree celcius? What wull be the length of rod at 0 degree celcius?
I want to see the solution
Solution
Asumption
α
z
=
26
×
1
0
−
6
/
°
C
α
z
=26×10
−6
/°C
α
s
=
11
×
1
0
−
6
/
°
C
α
s
=11×10
−6
/°C
So length of scale at 25°C
100
=
l
0
(
1
+
α
z
Δ
t
)
100=l
0
(1+α
z
Δt)
l
0
=
99.93
c
m
l
0
=99.93cm
So
Δ
l
Z
=
99.93
×
1
0
−
6
×
26
×
25
Δl
Z
=99.93×10
−6
×26×25
Δ
l
Z
=
0.064
c
m
Δl
Z
=0.064cm
So lenght of rod at 25°c is
l
r
o
d
l
rod
At 25°C =100.064cm
Actual lenght of rod is
l
r
o
d
(
m
e
a
s
u
r
e
d
)
=
l
a
c
t
u
a
l
(
1
+
(
(
α
Z
n
−
α
r
o
d
)
Δ
t
)
l
rod (measured)
=l
actual
(1+((α
Zn
−α
rod
)Δt)
To determine the true length of the aluminum rod at 25 degrees Celsius, we need to take into account its thermal expansion. Aluminum expands when it is heated, which means the rod will be longer at 25 degrees Celsius compared to its length at 0 degrees Celsius.
To calculate the true length of the rod at 25 degrees Celsius, we can use the principle of linear thermal expansion:
ΔL = α * L * ΔT
Where:
ΔL is the change in length of the rod,
α is the coefficient of linear expansion for aluminum,
L is the original length of the rod, and
ΔT is the change in temperature.
The coefficient of linear expansion for aluminum is typically around 0.000022 per degree Celsius.
Given that the rod appears to be 1 meter long at 25 degrees Celsius and that the scale is correct at 0 degrees Celsius, we can calculate the true length of the rod at 25 degrees Celsius:
ΔL = α * L * ΔT
ΔL = 0.000022 * 1 * (25 - 0) = 0.00055 meters
Therefore, the true length of the rod at 25 degrees Celsius is 1 meter + 0.00055 meters, which equals approximately 1.00055 meters.
To determine the length of the rod at 0 degrees Celsius, we can use the same equation, but with the change in temperature being from 25 degrees Celsius to 0 degrees Celsius:
ΔL = α * L * ΔT
ΔL = 0.000022 * 1.00055 * (0 - 25) = -0.00055125 meters
Here, we get a negative value for the change in length, indicating that the rod contracts when cooled. Therefore, the length of the rod at 0 degrees Celsius would be 1 meter - 0.00055125 meters, which equals approximately 0.99944875 meters or roughly 0.99945 meters.