Mass Moment of Inertia: Capsule

MMOI of a Capsule

In this post we calculate the mmoi of a capsule.

To calculate the mmoi of a capsule, we consider it as the sum of a rectangle and two semi-circles. Then we can add the mmoi of each of the components and add them together.

We already know the mmoi of a rectangle, around the z-axis through its centre of mass.

But what of a semi-circle? It is in fact, half the mmoi of a full circle - if this is not obvious, go back to how we derived the mmoi of a full circle in the previous circle, consider a semi-circle as consisting of very thin semi-circular rings. The "length" of a semi-circular ring is obviously half the length of a full "hoop". The rest of the derivation is the same - hence the mmoi is:

Then we can use the parallel axis theorem to add the mmoi of the elements together to arrive at the I of the total.

And that is quite enough algebra for one post.

Javascript Implementation

  calculateMass(density = 0.005) {
  
    // the mass of a capsule is the mass of the rectangular section plus the mass
		// of two half circles (really just one circle)
    const h = this.height;
    const w = Phaser.Geom.Line.Length(this); // w is the distance between the centres of the end circles
    const r2 = this.radius * this.radius;
    
    // compute the rectangular area
    let ra = this.height * w ; // area of the "rectangle";
    let rm = ra * density; // mass of the rectangle area
    
    let ca = Math.PI * r2 ; // area of the "circle area"
    let cm = ca * density ; // mass of the circle area
    
    this.area = ra + ca ; // total area of the capsule
    this.mass = rm + cm;  // total mass
    this.inv_m  = this.mass === 0 ? 0 : 1 / this.mass;
    // inertia is the rectangular region plus the inertia of the 2 half circles at a distance from centre of mass
    let d = w * 0.5; // distance of the centre of the "circle" from the centre of mass
    // parallel axis theorem I2 = Ic + m * d^2
    let cI = 0.5 * cm * r2 + cm * d * d;
    let rI = rm * (this.height * this.height + w * w) / 12;
    // add the rectangular inertia and cicle inertia
    this.inertia = cI + rI;
    this.inv_inertia = this.mass === 0 ? 0 : 1 / this.inertia;
   
  }