How to find the moment of inertia of a solid cone about its central axis..??

3 years ago

Share

Answers : (2)

                                        

2087_16302_moment of inertia of a cone about axis.jpg


 


Please approve !

3 years ago
                                        

Dear student,


The moment of inertia of a cone about its central axis, start with the standard Intertia equation


I = integral r^2 dm


dm = rho dV (rho is density) (dV is basically volume)


dV = r dr dtheta dx


not going to prove that here but you will see in the integral that this does indeed form the volume. integral will be refered to as int from here on.


This now forms the triple integral


I = rho int(0 to H) int(0 to 2pi) int(0 to r) r^3 dr dtheta dx


solving the integral leaves


I = rho int(0 to H) int(0-2pi) 1/4 r^4 dtheta dx


solving the second integral leaves


I = rho int(0 to H) 1/2 pi r^4 dx


ok so now you have to sub in the equation for r (the radius) of the cone


r = (R/H)x


this is the radius at the base divided by the height of the cone multiplied by the distance along the x axis. this equation gives you r at any point this gives you


I = rho int(0 to H) 1/2 pi [(R/H)x]^4 dx


time to do some housekeeping and take all the constants outside the integral


I = (rho pi R^4) / (2 H^4) int(0 to H) x^4 dx


this can now be solved and simplified to


I = (rho pi R^4 H) / 10


At this stage your solution is complete, however you can tidy up the equation by taking out the mass term.


m = (rho pi H R^2) / 3


split the Inertia term up to serperate out the mass term


I = [(rho pi H R^2) / 3]*[ (3R^2)/10 ]


this is now the complete solution in terms of mass


I = (3mR^2)/10



3 years ago

Post Your Answer

More Questions On IIT JEE Entrance Exam

Ask Experts

Have any Question? Ask Experts
Post Question
 
Answer ‘n’ Earn
Attractive Gift
Vouchers
To Win!!!
Click Here for details
Hello Sir..!! my friend is very intelligent. but her capacity to score is 920/1000 in AP .How much should she score in JEE examz to get into IIT??
 
 
she should focus in getting atleast 200+ marks in JEE mains....
  img
Amit Shekhar 2 months ago
 
200+ marks...
  img
Amit Shekhar 2 months ago
How Should I Plan My Daily Time Table ?
 
 
Hi student First count how much time you have for studies other than your daily routine work. Then divide the time according to your need of study and type of subject for study.Study hard...
  img
Anjali Ahuja 15 days ago
Find the locus of the mid point of the chord of a circle x 2 + y 2 = 4 such that the segment intercepted by the chord on the curve x 2 - 2x – 2y =0 subtends a right angle at the origin.
 
 
ohh , really?? have you even tried to draw the figure urself ??? all the segments intercepted on this curve inside the circle subtend an obtuse angle
 
abhidon3 24 days ago
 
well its a correct question
 
Vikrant Prasad 24 days ago
 
hey man ,, why you guyz post such a question which are either wong or incompete ,, it wastes too much time ,,,
 
abhidon3 25 days ago
For some reasons I will not be able to take classroom coaching for iit jee.So,i have enrolled in a postal course for 2 years.Is it enough to get me into iit(apart from my hardwork) or is...
 
 
Hi There are times during ur preparation when books alone cannot help,, u need some expert guidance to undertand some difficult concepts and tricks to excel in d xam. Classroom coaching...
 
AskiitianExpert Shine 5 years ago
what is the use of state rank in jee mains?
 
 
Hi, Satae ranks are used to fill the state quota seats in the NITs of a specific state. So if you are planning to take admission in ur home state then you state rank will be used. While AIR...
  img
Yash Baheti 4 months ago
sir this year i am giving ntse exam...but i am not taking classes for it...so sir kaise prepare karna hoga..
 
 
Most of thing will come from your school syllabus. First, you should focus on the school study, plus there is a book for NTSE exam by Tata Mcgraw hill. You should focus on the book as well....
  img
Jitender Singh 5 months ago
View all Questions »