The principal moments of inertia at the centroid C for the semicircular region shown are I1=81.43×106mm4 and I2=22.77×106mm4. And the principal directions are along x and y axes.
To determine
(b)
The moments and the products of inertia about the u-v-axes for the semicircular region shown.
Expert Solution
Answer to Problem 9.51P
Moments of inertia:
Iu=33.2×106mm4
Iv=71.0×106mm4
Products of inertia:
Iuv=22.5×106mm4
Explanation of Solution
Given information:
For the semicircular region shown:
I1=Iy=81.43×106mm4
I2=Ix=22.77×106mm4
Calculations:
12(Ix+Iy)=12(22.77+81.43)×106=52.10×106mm412(Ix−Iy)=12(22.77−81.43)×106=−29.33×106mm4Moments of inertia about the u-v-axes, using the relations:Iu=12(Ix+Iy)+12(Ix−Iy)cos2θ−Ixysin2θIu=[52.10−29.33cos(−50o)−0]×106⇒Iu=33.2×106mm4Iv=12(Ix+Iy)−12(Ix−Iy)cos2θ+Ixysin2θIv=[52.10+29.33cos(−50o)+0]×106⇒Iv=71.0×106mm4Hence, Products of inertia about the u-v-axes:Iuv=12(Ix−Iy)sin2θ+Ixycos2θIuv=[−29.33 sin(−50o)+0]×106⇒Iuv=22.5×106mm4
Conclusion:
For the semicircular region, the moments of inertia about the u-v axes are Iu=33.2×106mm4 and Iv=71.0×106mm4. And the products of inertia about the u-v axes is Iuv=22.5×106mm4.
Want to see more full solutions like this?
Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!
For the same case, consider the moment of inertia about each of the four axes. About which axis will the MoI be the smallest number?
Give me right solution according to the question
2. Locate the X-X and Y-Y centroidal axes, and calculate the moments of inertia with respect to both
centroidal axes for the areas shown below.
1
Inswer: (1) X:
(2) y:
(3) With respect to the X-X centroidal axis:
With respect to
the Y-Y centroidal axis:
in
in.
Chapter 9 Solutions
International Edition---engineering Mechanics: Statics, 4th Edition
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.