Suppose we have a metal bar of length L0 metres at temperature T0 in degrees Celsius.When the bar is heated or cooled, the length of the bar changes. The amount that the length changes is proportional to the product of the temperature change and the original length of the bar L0. Let á be the proportionality constant of this change where the units for á is measured in m/m◦C (metres per metres degree Celsius). Find an expression for the length L of the metal bar as a function of the temperature T.Draw a sketch of the graph of this function, labelling the point where T = 0."
I am having trouble coming up with the equation. How can I turn the proportionality equation described above to one with L as a function of T?
dL = a Lo dT
L = a Lo T + constant
when T = 0, L = Lo
so
L = (a Lo) T + Lo
To is y (or in this case L)axis intercept
(a Lo) is slope
Thanks for your help. I see the question much clearly now.
To find an expression for the length of the metal bar as a function of temperature, we can start with the proportionality relation given:
ΔL = α * ΔT * L0
where:
ΔL is the change in length of the bar,
α is the proportionality constant measuring the change per degree Celsius (m/m°C),
ΔT is the temperature change in degrees Celsius, and
L0 is the original length of the bar.
To convert this equation into a function of the length L with respect to temperature T, we need to integrate both sides. But first, we can simplify the equation by assuming that the change in length and temperature is small (ΔL and ΔT are small). Then we can use infinitesimal changes, denoted as dL and dT, respectively:
dL = α * dT * L0
Now, we can integrate both sides of the equation:
∫dL = ∫α * L0 * dT
The integral on the left side will simply be the function L, and the integral on the right side is just the product of α and L0 integrated with respect to T:
L = α * L0 * ∫dT
Since α and L0 are constants, we can factor them out of the integral:
L = α * L0 * ∫dT
L = α * L0 * (T + C)
Here, C is the constant of integration, which accounts for the initial temperature T0 when T = 0.
Therefore, the expression for the length L of the metal bar as a function of temperature T is:
L = α * L0 * (T + C)
To sketch the graph of this function, plot the length L on the y-axis and the temperature T on the x-axis. The graph will be a straight line with a positive slope of α * L0, indicating that as temperature increases, the length of the metal bar also increases. The point where T = 0 corresponds to the y-intercept, which is at L = α * L0 * C.