Moment Of Inertia Of A Semicircle

Moment of inertia of a semicircle is generally expressed with the following equation;

How To Find The Moment Of Inertia Of A Semicircle

In order to find the moment of inertia of a semicircle, we need to recall the concept of deriving the moment of inertia of a circle. The concept can be used to easily determine the moment of inertia of a semicircle.

1. We will first begin with recalling the expression for a full circle.

I = πr⁴ / 4

In order to find the moment of inertia, we have to take the results of a full circle and basically divide it by two to get the result for a semicircle.

Now, in a full circle because of complete symmetry and area distribution, the moment of inertia relative to the x-axis is the same as the y-axis.

I_x = I_y = ¼ πr⁴

M.O.I relative to the origin, J_o = I_x + I_y = ¼ πr⁴ + ¼ πr⁴ = ½ πr⁴

Now we need to pull out the area of a circle which gives us;

J_o = ½ (πr²) R²

Similarly, for a semicircle, the moment of inertia of the x-axis is equal to the y-axis. Here, the semi-circle rotating about an axis is symmetric and therefore we consider the values equal. Here the M.O.I will be half the moment of inertia of a full circle. Now this gives us;

I_x = I_y = ⅛ πr⁴ = ⅛ (A_o) R² = ⅛ (πr²) R²

Now to determine the semicircle’s moment of inertia we will take the sum of both the x and y-axis.

M.O.I relative to the origin, J_o = I_x + I_y = ⅛ πr⁴ + ⅛ πr⁴ = ¼ πr⁴

Derivation

We will basically follow the polar coordinate method.

The moment of inertia of the semicircle about the x-axis is

y = r sin θ

dA = r drd θ

We will now determine the first moment of inertia about the x-axis. We get;

I_x = _o∫^r_o∫^π y² ⋅ dA

= _o∫^r_o∫^π (r sin θ) ²  . r drd θ

= _o∫^r_o∫^π r³ [(1 – cos2θ)/2](1/2) d2θdr

We will now determine the first moment of inertia about the x-axis. We get;

I_y = _o∫^r_o∫^π x² ⋅ dA

x = r cosθ

dA = r drd θ

I_y = _o∫^r_o∫^π (r cos θ) ²  . r drd θ

= _o∫^r_o∫^π r³ [(1 + cos2θ)/2](1/2) d2θdr

J₀ = I_x + I_y

= (1/8) π r⁴ +(1/8) π r⁴

= (1/4) π r⁴

⇒ Check Other Object’s Moment of Inertia:

• Moment Of Inertia Of Circle
• Moment Of Inertia Of A Quarter Circle
• Moment Of Inertia Of Ring
• Moment Of Inertia Of A Sphere
• Moment Of Inertia Of A Disc

Parallel Axis Theorem