I do not understand what this question is asking and how to get the result for it:
"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."
So far I have figured that you have to have these values of length and temperature proportional to each other so like when length increases, the temperature decreases and as length decreases the temperature increases. But I do not know how to write that and how to come about drawing the graph.
Well, let's see what they have said.
dL = k dT L0
so,
L = k*L0*T + C
Now you can plug in initial conditions or boundary conditions to determine k and C
To solve the problem, you need to find an expression that describes how the length of the metal bar (L) changes with respect to the temperature (T).
According to the given information, the change in length (ΔL) is proportional to the product of the temperature change (ΔT) and the original length of the bar (L0). The proportionality constant for this relationship is represented by á, measured in m/m◦C.
So, we can write the equation as:
ΔL = á * ΔT * L0
Since we want to find the expression for the length L as a function of the temperature T, we need to integrate both sides of the equation. Integrating the left side of the equation will give us the expression for the change in length as a function of temperature, which we can then use to find the complete expression for length.
Integrating the right side of the equation is straightforward since the constants L0 and á are not variables of integration. We obtain:
∫ΔL = ∫(á * ΔT * L0)
∫dL = L = ∫(á * dT * L0)
L = á * L0 * ∫dT
Integrating ∫dT will simply give us T, so:
L = á * L0 * T
This equation represents the length of the bar (L) as a function of temperature (T). Now, let's draw a sketch of the graph of this function.
Since the expression for L is linearly dependent on T (L = á * L0 * T), the graph will be a straight line. The slope of the line is given by the proportionality constant á, and the y-intercept is zero when T = 0.
To draw the graph, you can take the values of T and calculate the corresponding values of L using the given equation. Plot these points and connect them to form a straight line. The line will start at the origin (0, 0) and extend in the direction determined by the sign of the constant á.
It's important to note that without explicit values for á, L0, or any specific temperature values, we cannot determine the exact shape, steepness, or position of the graph, but we can illustrate the general behavior based on the given information.