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Moment of Inertia of Hollow Cone

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Last updated date: 17th Jul 2024
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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 about 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;

 

\[I = \frac {MR^2}{2}\]

 

What is Inertia?

Simply explained, an object continues to be in motion or rest until an external source acts on it and this tends to be in motion or the rest of an object is called inertia. For example, as soon as the bus or a vehicle starts moving, all the passengers feel a backward push as they were in a state of rest until the external force, in this case, movement of the bus acted on them. The law of inertia also known as the first law of motion was coined by Sir Issac Newton. Inertia is further divided into the inertia of rest, inertia of motion and finally inertia of direction. 

 

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;  

\[\frac {r}{x} = \frac {R}{\sqrt {R^2 + H^2}}\]

\[r=\frac {R}{R} = \frac {R}{\sqrt {R^2 + H^2}}\]

\[r=\frac{\frac{R}{R}}{\sqrt {R^2 +H^2 \times x}}\]

The weight of the elementary disk mass is given by;

dm = \[\frac {M}{\pi R \sqrt {R^2 +H^2 \times \pi rdx} }\]

dm= \[\frac{\frac{M}{\pi R \sqrt {R^2 + H^2 \times 2 \times R}}}{R \sqrt {R^2 + H^2 \times xdx}}\]

dm =  \[\frac {2Mxdx}{R^2 + H^2}\]

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

dI = r2 dm

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

  1. Finding the moment of inertia through integration.

I = \[\frac{\frac{\int 2 mx dx}{R^2 + H^2 \times R^2}}{R^2 + H^2 \times x^2}\]

I = \[\frac {2MR^2}{(R^2 + H^2)^2} \int x^3 dx\] 

Using the limits of x where x usually varies from o to \[\sqrt {R^2 + H^2}\]

We now get;  

I = \[\frac{\frac{2MR^2}{(R^2 + H^2)^2 \times (R^2 + H^2)^2}}{4}\]

I = \[\frac {1}{2} MR^2\]

Thus, \[I = \frac {MR^2}{2}\]

 

Tips to improve scores in Physics, IIT JEE:

Physics is one of the three subjects included in IIT JEE, the others being, math and chemistry. It might be difficult for students to remember all the theorems, formulas, laws and other concepts but, with proper strategy and some exam tips, one can score 90+ marks in physics. Following are some such tips that would help students:

 

Rather than looking at the solutions as soon as you see a problem, look at the question and try to solve them on your own. Then check your answer with the solutions given. Try to observe how the problem was solved in the solutions given and learn different easy ways to solve a question. This will help in improving your answer solving skills and also teaches you the easiest method to solve a problem.

 

Some particular chapters in physics have comparatively more weightage than the others. Try to concentrate on them more. However, you are not supposed to leave any chapter or topic completely. Try to finish as many portions as you can and revise them according to the weightage. These important questions and weightage can be accessed from Vedantu.

 

Try to finish the daily portion and solve DPP sheets on time. If you still feel that you are lagging, join an online test series to solve more and more problems and learn from them. Do not forget that revision is the key point to score better than other students.

FAQs on Moment of Inertia of Hollow Cone

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

A hollow cone is a geometrical shape that looks similar to a pyramid but is hollow from the inside. The moment of inertia of any given hollow cone is determined by the formula  \[I = \frac {MR^2}{2}\]. Here, “I” denotes inertia, “R” denotes the radius of the hollow cone and “M” denotes the mass of the 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?

A hollow cone is a geometrical figure that looks similar to a normal cone or pyramid from the exterior but hollow on the inner side. 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 \[\frac {2}{3}\] of the way from the pole to the base midpoint. To know clearly about the formula derivation and application, refer to the above article. 

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. A hollow cone is a geometrical figure which is quite similar to a normal cone or a pyramid from the outside but is hollow from inside. A hollow cone has an infinite number of triangles on the external side. 

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

A hollow cone is a geometrical figure that looks similar to a pyramid or a normal cone from the outside but it is hollow on the inside. The Center of mass is the point of a figure where the entire mass of the figure concentrates and it is very important to balance the object. 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 \[\frac {3}{4}\] of the height of the cone. That implies that the center of mass is 1/4 of the base height. 

5. What is the difference between the center of mass, moment of inertia and volume of a hollow cone and how do you find the volume of a hollow cone?

A hollow cone is a geometrical figure that looks similar to a pyramid or a cone from the outside but is hollow from the inside. The Center of mass of any object is the point at which the entire mass of the object concentrates and it is an important point to balance the object. For a hollow cone, the center of mass is \[\frac {1}{4}\] th of base height. The volume of an object is the space occupied by the object measured in 3 dimensions and the volume of a hollow cone is measured by the formula, “\[(\frac {1}{4})\] * π * r² * h”. A moment of inertia is nothing but a measure of the rotational inertia of a body. In the above formulas, r is the radius of the cone and h is the height of the cone. 


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