Moment of Inertia of Hollow Cone

Introduction: Moment of Inertia

In science, a moment of inertia, quantitative measure of a body's rotational inertia— that is, the reaction that the body demonstrates to getting its rotational speed around an axis altered by applying a torque (turning force). The axis can be internal or external and can be fixed or not.

However, the moment of inertia (I) is always specified in relation to that axis, and is defined as the sum of the products obtained by multiplying the mass of each particle of matter in a given body by the square of its distance from the axis. When determining angular momentum for a rigid body the moment of inertia is equal to the mass at linear momentum.

The momentum p is equal to the mass m times the velocity v for linear momentum; whereas for angular momentum, the angular momentum L is equal to the moment of inertia I times the angular velocity.

The moment of hollow cone inertia can be determined using the expression below;

I = MR2 / 2

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Moment of Inertia of Hollow Cone Formula Derivation

We should basically follow certain general guidelines which are to extract the moment of inertia formula of a hollow cone

  • Determining the mass density per unit area.

  • Defines parameters and discovers the mass of small parts.

  • Consider the moment of inertia corresponding to a ring for the small parts.

1. A hollow cone with radius R, height H, and mass M is located.

Now, at a slant height l and radius r having thickness dl and mass dm, we'll take the product disk.

Using the similarity of the triangle we get;

r / x = R / √ R2 + H2

r = R / R / √ R2 + H2 X x

The weight of the elementary disk mass is given by;

dm = M / π R√ R2 + H2 X 2 π rdx

dm = M / π R√ R2 + H2 X 2 X R / R√ R2 + H2 X xdx

dm = 2Mxdx / R2 + H2

 

2. Calculating the time of elementary disk inertia. It is given as;

dI = r2 dm

dI = 2Mxdx / R2 + H2 X R2 / R2 + H2X x2

3. Finding the moment of inertia through integration.

I = ∫ 2Mxdx / R2 + H2 X R2 / R2 + H2X x2dx

I = 2MR2 / (R2 + H2)2 ∫ x3dx

Using the limits of x where x usually varies from o to √ R2 + H2

We now get;

I = 2MR2 / (R2 + H2)2 X (R2 + H2)2 / 4

I = ½ MR2

Thus, I = MR2 / 2

FAQ (Frequently Asked Questions)

1. What is the Moment of Inertia of a Hollow Cone?

Let's go in! A Hollow Cone's Inertia Moment. A disk about its axis has rotational inertia of 0.7kgm2.

2. What is the Centre of the Mass of a Hollow Cone?

The surface of a hollow cone may be considered to consist of an infinite number of triangles of infinitesimally slender isosceles, and thus the center of mass of a hollow cone (without foundation) is 2/3 of the way from the pole to the base midpoint.

3. What is a Hollow Cone?

A hollow cone spray pattern consists of droplets centered on a ring-shaped region of effect, with no droplets dropping inside the conic diameter. Hollow cone nozzles create a very fine atomized liquid mist under the same operating conditions and can collect a higher rate of suspended particles than other nozzles.

4. Where is the Center of Mass of a Cone?

The middle of a tip's mass lies along a line. This line crosses the peak and is perpendicular to the surface. The center of mass with respect to the apex is a point 3/4 of the height of the cone. That implies that the center of mass is 1/4 of the base height.