A uniform plate of height = 1.80 m is cut in the form of a parabolic section. The lower boundary of the plate is defined by: = 1.20 . The plate has a mass of 1.29 kg. Find the moment of inertia of the plate about the y-axis.
You need to clarify this statement:
<The lower boundary of the plate is defined by: = 1.20 . >
and somehow establish the width X of the parabola at the open end.
The moment of inertia will be some fraction of M X^2 that can be established by calculus.
it is defined by y=1.20x^2.
To find the moment of inertia of the plate about the y-axis, we need to integrate the differential element of the plate's mass with respect to the y-axis squared.
The moment of inertia (I) is given by the integral of (r^2 * dm), where r is the perpendicular distance from the axis of rotation to the differential mass element (dm).
In this case, we have a parabolic section defined by the equation y = x^2/a, where a is a constant. However, in order to integrate with respect to y-axis, we need to express x in terms of y.
From the equation of the parabola, x = sqrt(a * y). Now we can express the differential mass element (dm) in terms of y.
The height (h) of the plate is given as 1.80 m, and the lower boundary is y = 1.20 m. So, the upper boundary of the plate will be y = 1.80 m.
Now, let's find the mass of the plate per unit area. Mass per unit area (m/A) is given by the total mass (m) divided by the area (A) of the plate.
m/A = m / (a * h)
Given that the mass (m) is 1.29 kg and the height (h) is 1.80 m, we can substitute these values into the equation and calculate m/A.
m/A = 1.29 kg / (a * 1.80 m)
Now, we can express dm in terms of dy (since we are integrating with respect to the y-axis). dm = (m/A) * dy
Substituting the value of m/A, we get:
dm = (1.29 kg / (a * 1.80 m)) * dy
Next, we need to calculate r^2, which is the square of the perpendicular distance from the y-axis to the differential mass element.
r^2 = (sqrt(a * y))^2
= a * y
Now, we can substitute the values of dm and r^2 into the equation for moment of inertia.
I = ∫(r^2 * dm)
= ∫((a * y) * (1.29 kg / (a * 1.80 m)) * dy)
Integrating this expression from the lower boundary y = 1.20 m to the upper boundary y = 1.80 m will give us the moment of inertia of the plate about the y-axis.