Derive the expression for the moment of inertia of a cylinder length T.radius r and density *w* about longitudinal centroidal axis and about the centroidal transverse axis
Answers
Answer:
We will consider the cylinder having mass M, radius R, length L and the z-axis which passes through the central axis.
Here,
Density ρ = M / V
Next, we will consider the moment of inertia of the infinitesimally thin disks with thickness dz.
First, we assume that dm is the mass of each disk, We get;
dm = ρ x Volume of disk
dm = (M / V) x (πr2.dz)
We take V = area of circular face x length which is ( πr2L).
Now we obtain;
dm = (M / πr2L) x (πr2.dz)
dm = (M / L) dz
The moment of inertia about the central axis is given as;
dlz = ½ dmR2
2. Use of Perpendicular Axis Theorem
We now apply the perpendicular axis theorem which gives us;
dlz = dlx + dly
Here, if we need to consider that both x and y moments of inertia are equal by symmetry.
dlx = dly
We need to combine the equations for the perpendicular axis theorem and symmetry. We get;
dlx = dlz / 2
Now we substitute lz from the equation above.
dlx = ½ x [½ dmR2]
dlx = ¼ dmR2
Alternatively, for the x-axis, we use the parallel axis theorem to find the moment of inertia. We get;
dlx = ¼ dmR2 + dmz2
3. Integration
Now we conduct integration over the length of the cylinder to express the mass element dm in terms of z. We take the integral from z=0 to z=L.
Ix = o∫L dlx
Ix = o∫L ¼ (M / L) R2 dz+ o∫L z2 (M / L)dz
Ix = [¼ (M / L) R2z + (M / L) z3 / 3]0L
Since it is a definite form of integral we ignore the constant. We will now have;
Ix = ¼ (M / L) R2 L + ML3 / 3
Ix = MR2 /4+ ML2/3
⇒ Check Other Object’s Moment of Inertia:
Moment Of Inertia Of A Solid Cylinder
Moment Of Inertia Of A Hollow Cylinder
Moment Of Inertia Of A Rectangular Plate
Moment Of Inertia Of Rectangle
Moment Of Inertia Of Rod
Test Your Knowledge On Moment Of Inertia Of A Cylinder!
Quiz Image
Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin!
Select the correct answer and click on the “Finish” button
Check your score and answers at the end of the quiz
Type your search
BNAT
BNAT
JEE Main 2021 Question Papers With Solutions
Enroll for a FREE 60 minute demo class today
Daily Quiz Videos
JEE Preparation 2020
NEET Preparation 2020
Ask A Doubt
Join BYJU'S Learning Program
Name
Mobile Number
City / Town
Grade/Exam
Email Address
COURSES
CBSE
ICSE
CAT
IAS
JEE
NEET
Commerce
JEE Main
NCERT
JEE Advanced
CBSE Sample Papers
CBSE Question Papers
EXAMS
CAT Exam
GATE Exam
IAS Exam
UPSC Syllabus
UPSC 2021
Bank Exam
Government Exams
CLASSES
Class 1st - 3rd
Class 4th - 5th
Class 6th - 10th
Class 11th - 12th
EXAM PREPARATION
Free CAT Prep
Free IAS Prep
Maths
Ph
Explanation: