HOLLOW CONE

We have defined some characteristic points and some parameters to conveniently see the motion and energy dynamics of a body. In the same sequence, we will be learning about such parameters for a hollow cone. The centre of mass is a point where all of the mass of a body is assumed to be concentrated. E.g. The calculated centre of mass of a hollow cone should only be a term of mathematical importance to us. But we will also see that following this assumption can put us in some trouble if we are going to follow them blindly. There are also terms that are there to let us visualise the analogy between translational motion and vibrational motion.

CENTRE OF MASS OF HOLLOW CONE

Let us imagine that there is a hollow cone of mass M. The radius of the base of the cone is R. The height of the cone is H. Let us assume the slant height to be L and the variable distances in these directions are the same in the small case i.e. r,h and l. Let us also introduce a term J which is the surface mass density and the semi-vertical angle to be θ.

J= M/𝝿RL

Now, we define the coordinates of the centre of mass of a hollow cone or simply any 2D or 3D shape as;

XCM=∫xdm/M ; where the x is the x coordinate of the corresponding to the dm mass

YCM=∫ydm/M ; where the y is the y coordinate of the corresponding to the dm mass

ZCM=∫zdm/M ; where the z is the z coordinate of corresponding to the dm mass

From the symmetry of a right circular cone, we can say that the x coordinate of the centre of mass of the hollow cone is x=0.

Now, our whole concentration is to look for the y coordinate of the centre of mass of the hollow cone. 

YCM=∫ydm/M

YCM=∫yJ2𝝿xdl/M 

     =0H∫y²2tanθsecθdy/J𝝿RL

    = 2/3H

MOMENT OF INERTIA OF HOLLOW CONE

Moment of inertia is a term in rotational dynamics to see the mass analogy with the translational or the 1D motion. For a more clear view of the scenario let’s have a look at the following formula

torque= I𝞪; where 𝞪 is the angular acceleration of the body which you can set analogy with newton’s second law of motion ie. F= ma 

So you will get to see the analogy that for the same mass more force was needed to be applied. In the same way, if there are two bodies with a different moment of inertia, the one with less moment of inertia will have a greater angular acceleration than the one with a higher moment of inertia for the same applied torque.

We will derive the expression for the moment of inertia of a hollow cone in the same way by taking a small element and integrating it like the way we did for the centre of mass of a hollow cone.

We define the moment of inertia about a line. Mathematically we can write the moment of inertia is;

I=∫r²dm; where the r us the distance between the mass dm and the kine about which we are defining the moment of inertia.

Let us derive the expression for the moment of inertia of the hollow cone about the axis of the right circular cone.

I=∫x²dm

=∫x²J2𝝿xdl

=∫x²J2𝝿xcosecθdx     ; as l=xcosecθ

=MR²/2

CENTRE OF GRAVITY OF HOLLOW CONE

Before getting into the further discussion of the centre of gravity of hollow cones, let us see what is a centre of gravity and how and when it will be different from the centre of mass. The Centre of gravity is the point where the resultant of all the weights of the system act. Moreover, the moment of gravity about the centre of gravity is always equal to zero.

So it is a matter of concern when the centre of mass and the centre of gravity of a hollow cone will differ?

For a body of small dimension, as compared to the earth that the gravitational force is uniform all about the body, the centre of gravity and the centre of mass of hollow cones coincide and will be at the geometrical centre of the body. And they will be at different positions if the body is subjected to a non-uniform gravitational field such that the gravity is different at the different points of the body.

The way the centre of mass shifts towards the direction more mass is concentrated, the centre of gravity also shifts towards the direction where the gravitational force is higher i.e. the weight is more.

CONCLUSION

We come through many expressions like moment of inertia of hollow cone, centre of mass of hollow cone and their importance and mathematical significance. But we should keep in mind that the centre of mass is something of mathematical significance only. These terms were introduced to let the learners understand the analogy of the complex dynamics with the simpler ones. For example, the moment of inertia can be seen as a rotational analogue of the mass. In the same way we can depict the motion of the complex bodies as a motion of a point where the whole mass of the body is concentrated. But we can assume these things only when we come through their derivation.