Question

Explain thearem of parallel ares​

Answer 1

Explanation:

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Answer 2

Answer:

What is Parallel Axis Theorem?

Parallel axis theorem states that

The moment of inertia of a body about an axis parallel to the body passing through its center is equal to the sum of moment of inertia of body about the axis passing through the center and product of mass of the body times the square of distance between the two axes.

Parallel Axis Theorem Formula

Parallel axis theorem statement can be expressed as follows:

I = Ic + Mh2

Where,

I is the moment of inertia of the body

Ic is the moment of inertia about the center

M is the mass of the body

h2 is the square of the distance between the two axes

Parallel Axis Theorem Derivation

Let Ic be the moment of inertia of an axis which is passing through the center of mass (AB from the figure) and I be the moment of inertia about the axis A’B’ at a distance of h.

Consider a particle of mass m at a distance r from the center of gravity of the body.

Then,

Distance from A’B’ = r + h

I = ∑m (r + h)2

I = ∑m (r2 + h2 + 2rh)

I = ∑mr2 + ∑mh2 + ∑2rh

I = Ic + h2∑m + 2h∑mr

I = Ic + Mh2 + 0

I = Ic + Mh2

Hence, the above is the formula of parallel axis theorem.

Parallel Axis Theorem of Rod

The parallel axis theorem of rod can be determined by finding the moment of inertia of rod.

Moment of inertia of rod is given as:

I = 13 ML2

The distance between the end of the rod and its center is given as:

h = L2

Therefore, the parallel axis theorem of rod is:

Ic = 13ML2 – ML22

Ic = 13ML2 – 14ML2

Ic = 112 ML2

Perpendicular Axis Theorem

Perpendicular axis theorem states that

For any plane body the moment of inertia about any of its axes which are perpendicular to the plane is equal to the sum of the moment of inertia about any two perpendicular axes in the plane of the body which intersect the first axis in the plane.

Perpendicular Axis Theorem Formula

Perpendicular axis theorem is used when the body is symmetric in shape about two out of the three axes if the moment of inertia about two of the axes are known the moment of inertia about the third axis can be found using the expression:

Ia=Ib+Ic