SOLID CYLINDER

 In mathematics, a three-dimensional solid with an infinite curvilinear surface is referred to as a cylinder. In a solid cylinder, the parts of the line determined by the cylindrical surface element between two parallel planes are called the cylinder element and each of them has an equal length. A circuit bounded by a cylindrical surface on any parallel plane is called the cylinder base. The two foundations of the cylinder are parallel figures. 

The part of the line that connects the centers of the circular foundations is called the cylinder axis. The distance between the axis to the outer curve is called the radius of the cylinder and the vertical distance between the two corresponding circular bases is called the length (h) of the cylinder.

The shape of the cylinder:

The cylinder has two flat ends in a circular pattern. These two faces are connected by a curved face that resembles a tube. When you make a flat cylinder net, it looks like a rectangular circle attached to each end.

Cylindrical surface:

In some geometric and topology areas, the word cylinder refers to what has been called the cylindrical surface. The cylinder is defined as the area that connects all points to all lines corresponding to a given line and exceeds the curve of a fixed plane in a plane that does not correspond to a given line. Such cylinders, at times, have been called conventional cylinders. At each point of the standard cylinder passes a unique line contained in the cylinder.

An ellipse, parabola or hyperbola cylinder is called an elliptic cylinder, a parabolic cylinder and a hyperbolic cylinder, respectively. These are damaged quadric areas.

Moment of Inertia:

Moment of inertia is a physical property because it withstands angular acceleration, which is the sum of the products of the weight of each particle between the square of the distance from its axis rotation to the body. The formula for calculating the Moment of Inertia of a solid surface of mass M and radius R is as follows:

 I = ½ MR²

Derivation: 

Before starting derivation, let us consider a few things about solid cylinder: 

• Cut or split the solid cylinder into infinitesimal rings.
• Moments of infinitesimally thin cylindrical shells are needed to be summed up.
• The length of the ring is L along with the thickness dr. 

To calculate the moment of Inertia about its center, let’s take a solid cylinder of mass M, length L and radius R. 

Step 1: General equation of moment of Inertia

We know, 

dl = r²dm 

where, dm = p dV

To get the value of dm, we firstly need to find out the dV;

dV=dA L

in which, dA = area of big ring – area of small ring. Hence, 

 dA = π (r +dr)² – πr²

 dA = π (r² +2rdr + (dr)²)- πr ²

 Here, (dr)2 = 0 

 dA= 2πrdr

 Step 2: Put the of dA into dV

 dV= dAL = 2πrdrL  

By substituting the value of dV into dm, we will get; 

 dm=2πrdrLρ

Put the value of dm in dI, we get;

dI = ∫r²  (2πrdr)Lρ 

I = 2π Lρ R∫0 r³ dr

 I = 2πLρ [R4  /4] 

 Step 3: To find out the expression of density

  ρ=M ⁄ V

Then, 

   ρ = M/ πR² L

 Step 4: Use the polynomial integration  method to find out moment of Inertia of solid

I = 2πL M/ πR² L [R4 /4]

By solving, 

I = ½  MR²

The surface area of a cylinder: 

A three-dimensional shape is the area present at top of the cylinder. A cylinder is a three-dimensional structure with interlocking circular foundations. It has no vertices. Generally, a surface area is represented by a three-dimensional structure. The surface area is represented by square units. For example, cm², m² and so on. A set of circular discs packed with one another can exactly look like a kind of cylinder. A solid three-dimensional cylinder is both more spacious and  has enough volume.

Surface Area of Cylinder = Curved surface + Area of circular bases

Types of surface Area of Cylinder

1. Curved surface area
2. Total surface area

Curved surface Area: It is also known as Lateral Surface Area. The curved surface area of a cylinder is defined as the area of its curved area or area obtained after excluding the location of its two circular foundations.

Total surface area of a solid cylinder

The sum of the curved area and circular areas of the cylinder with radius ‘r’ and the length ‘l’ is the sum of the total surface area of the cylinder.

Calculation of total surface area of a solid cylinder: 

Total Surface Area = 2π × r × h + 2πr²

Or 

TSA = 2πr (r + h) 

                    (in square units) 

Conclusion:

A solid cylindrical band with two parallel planes is called a solid cylinder. Moment of inertia is the amount of energy needed to move an object. The value can be adjusted to increase or decrease the net. In sports such as ice skating, diving and gymnastics, athletes are constantly changing the shape of their bodies.