1. Topic : Moment of Inertia
PARUL INSTITUTE OF ENGINEERING & TECHNOLOGY
CIVIL DEPARTMENT
Topic : Moment of Inertia

2. Name of Students Hathila Jay : 130370106052
Parmar Shitul :
Pargi Bharat :
4. Parmar Ronak :
5.Mal pankaj:
6.Vasaiya Priyakant:

3. Moment of Inertia Defined
The moment of inertia measures the resistance to a change in rotation.
Change in rotation from torque
Moment of inertia I = mr2 for a single mass
The total moment of inertia is due to the sum of masses at a distance from the axis of rotation.

4. Two Spheres
A spun baton has a moment of inertia due to each separate mass.
I = mr2 + mr2 = 2mr2
If it spins around one end, only the far mass counts.
I = m(2r)2 = 4mr2 m m r

5. Mass at a Radius
Extended objects can be treated as a sum of small masses.
A straight rod (M) is a set of identical masses Dm.
The total moment of inertia is
Each mass element contributes
The sum becomes an integral
distance r to r+Dr
length L
axis

6. Rigid Body Rotation
The moments of inertia for many shapes can found by integration.
Ring or hollow cylinder: I = MR2
Solid cylinder: I = (1/2) MR2
Hollow sphere: I = (2/3) MR2
Solid sphere: I = (2/5) MR2

7. Playground Ride
A child of 180 N sits at the edge of a merry-go-round with radius 2.0 m and mass 160 kg.
What is the moment of inertia, including the child?
Assume the merry-go-round is a disk.
Id = (1/2)Mr2 = 320 kg m2
Treat the child as a point mass.
W = mg, m = W/g = 18 kg.
Ic = mr2 = 72 kg m2
The total moment of inertia is the sum.
I = Id + Ic = 390 kg m2 m M r

8. Parallel Axis Theorem
Some objects don’t rotate about the axis at the center of mass.
The moment of inertia for a rod about its center of mass:
Some objects don’t rotate about the axis at the center of mass.
The moment of inertia depends on the distance between axes.
h = R/2 M
axis

9. Perpendicular Axis Theorem
For flat objects the rotational moment of inertia of the axes in the plane is related to the moment of inertia perpendicular to the plane.
Iy = (1/12) Ma2 b
Ix = (1/12) Mb2 M a
Iz = (1/12) M(a2 + b2)

10. Spinning Coin
What is the moment of inertia of a coin of mass M and radius R spinning on one edge?
The moment of inertia of a spinning disk perpendicular to the plane is known.
Id = (1/2) MR2
The disk has two equal axes in the plane.
The perpendicular axis theorem links these.
Id = Ie + Ie = (1/2) MR2
Ie = (1/4) MR2 R M M R Id Ie
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