Respuesta :
Answer:
B = -0.215 s⁻¹, A₀ = 1.537º
Explanation:
To be able to make this graph we must linearize the data, the best procedure is to calculate the logarithm of the values
A = A₀ [tex]e^{-Bt}[/tex] + C
The constant creates a uniform displacement of the graph if we start the graph at the angle of 1, the constant disappears, this is done by subtracting 1 from each angle, so the equation is
(A-1) = A₀ e^{-Bt}
We do the logarithm
Log (A-1) = log Ao –Bt log e
Make this graph because paper commercially comes in logarithm 10, if we use graph paper if we can calculate directly in base logarithm e, let's perform this calculation
Ln (A-1) = Ln A₀ - Bt
To graph the points we subtract 1 from each angle and calculate the logarithm, the data to be plotted are
θ’= ln (θ -1)
θ(º) t(s) θ'(º)
6.2 2 1.6487
4.7 4 1.3083
3.2 6 0.7885
2.4 8 0.3365
1.8 10 -0.2231
We can use a graph paper and graph on the axis and the primary angle (θ') and on the x-axis time, mark the points and this graph is a straight line, we see that the point for greater time has a linearity deviation , so we will use the first three for the calculations
To find the line described by the equation
y -y₀ = m (x -x₀)
m = (y₂ -y₁) / (x₂ -x₁)
Where m is the slope of the graph e (x₀, y₀) is any point, let's start as the first point of the series
(x₀, y₀) = (2, 1.65)
(x₂, y₂) = (2, 1.65)
(x₁, y₁) = (6, 0.789)
We use the slope equation
m = (1.65 - 0.789) / (2-6)
m = -0.215
The equation is
y - 1.65 = -0.215 (x- 2)
y = -0.215 x +0.4301
We buy the two equations and see that the slope is the constant B
B = -0.215 s⁻¹
The independent term is
b = ln A₀
A₀ = [tex]e^{b}[/tex]
A₀ = [tex]e^{0.43}[/tex]
A₀ = 1.537º
