Center of Gravity and Moment of Inertia

Contents

**Define the term Centroid and Center of Gravity** 1

**Describe methods used for determination of Center of Gravity** 1

**List the C.G of simply geometrical figures** 2

**Recall different methods of Moment of Inertia** 5

**Describe the Perpendicular-axis theorem** 6

**Describe the Parallel-axis theorem** 7

**Define the term Centroid and Center of Gravity**

The term Centroid and Center of Gravity are related concepts in mechanics.

The Centroid is defined as the center of mass of a 2D object or a cross-sectional shape. It is the point where an object’s weight is evenly balanced and is represented by the geometric center of the shape. The centroid of a shape can be found by dividing the shape into smaller shapes and finding the center of mass of each shape and then finding the average of all the centres of mass.

The Center of Gravity (CoG), on the other hand, is defined as the point at which the weight of an object is considered to act. It is the point where the gravitational force on an object is considered to be concentrated. The CoG of an object is the point at which an object would balance if it were suspended. The location of the CoG of an object depends on its mass distribution, and it is always directly below the centroid.

In summary, the Centroid is the center of mass of an object, while the Center of Gravity is the point where an object’s weight is considered to act, and it is always below the centroid.

**Describe methods used for determination of Center of Gravity**

The Center of Gravity (COG) is a very important concept in Mechanics and it is the point where the entire weight of a body acts, as if it was concentrated at this point. It is also known as the center of mass, where the entire mass of an object is considered to be concentrated at a single point. The COG of a body plays a crucial role in understanding the behavior of a body when it is subjected to external forces and torques.

There are several methods used to determine the center of gravity of a body, which include:

- The graphical method: In this method, the body is represented on a graph and the lines are drawn from the body’s vertices to its center of gravity. The intersecting point of these lines represents the center of gravity.
- The analytical method: In this method, the body is divided into several parts, and the center of gravity of each part is calculated. Finally, the center of gravity of the whole body is calculated by taking the weighted average of the centres of gravity of each part.
- The experimental method: In this method, the body is suspended from different points and the point at which the body comes to rest is the center of gravity.
- The principle of moments: In this method, the moments of the weights of the different parts of the body about a reference point are calculated and the center of gravity is determined by equating the sum of the moments to zero.

Regardless of the method used, the determination of the center of gravity of a body is essential for analyzing the stability and motion of objects, as well as for determining the forces and torques acting on them.

**List the C.G of simply geometrical figures**

The center of gravity (C.G) of a simply geometrical figure refers to the point at which the entire weight of the figure is considered to be concentrated. The center of gravity is the point where the entire weight of the object is evenly distributed, making it the central point of balance. The C.G is an important concept in mechanics, as it determines the stability of an object and the direction in which it will fall when subjected to an external force.

The center of gravity of various simply geometrical figures can be determined using various methods, including:

- Rectangular figures: The C.G of a rectangular figure is located at its geometric center. The x-coordinate is equal to the average of the x-coordinates of its sides, and the y-coordinate is equal to the average of the y-coordinates of its sides.
- Triangular figures: The C.G of a triangular figure is located at its geometric center. It is the point where the three medians of the triangle intersect.
- Circular figures: The C.G of a circular figure is located at its geometric center. It is the point equidistant from all points on the circumference of the circle.
- Trapezoidal figures: The C.G of a trapezoidal figure is located at its geometric center. It is the point at which the line of action of the weight of the figure passes through.

It is important to note that these locations for the C.G of simply geometrical figures hold true only if the figures are made of uniform and homogeneous materials. If the figures are made of different materials, the C.G of each material must be determined and then combined to find the overall C.G of the figure.

**Describe and calculate the C.G of the following sections/bodies i. Symmetrical section ii. Unsymmetrical section iii. Solid Bodies iv. Cut-out section**

The Centroid (C.G) is the geometric center of a section or a body and is the point at which the body is in balance. The center of gravity (C.G) is the point where the weight of an object is concentrated and is considered to act. The C.G is a crucial aspect of mechanics as it determines the stability and balance of an object.

To determine the center of gravity, there are various methods that can be used:

- The Principle of Moments: By balancing the moments of weight on both sides of the C.G, the C.G can be determined.
- The Principle of Equilibrium: The C.G is the point where the object is in a state of static equilibrium, meaning that the sum of all forces and torques acting on the object is equal to zero.
- Geometrical Method: The C.G can be determined by dividing the section into smaller parts and finding the C.G of each part, then adding the C.G of each part to find the C.G of the entire section.

The C.G of simply geometrical figures such as a rectangle, a triangle, and a circle can be calculated using mathematical formulas.

The C.G of a symmetrical section can be found by taking the midpoint of the section. The C.G of an unsymmetrical section can be determined using the geometrical method.

The C.G of a solid body can be determined by dividing the body into smaller parts and finding the C.G of each part, then adding the C.G of each part to find the C.G of the entire body.

The C.G of a cut-out section can be determined by finding the C.G of the original section and then subtracting the C.G of the cut-out part.

**Describe Moment of Inertia**

Moment of Inertia, also known as the second moment of area, is a geometrical property of a cross-section or shape that defines the distribution of mass and its resistance to rotational motion around a given axis. It determines how much torque is needed to produce a given angular acceleration about a given axis.

In simpler terms, Moment of Inertia represents the measure of the distribution of mass in a body with respect to the axis of rotation, which is used to calculate the amount of rotational force required to cause rotational motion about the given axis. It is a scalar quantity, denoted by “I”. The unit of moment of inertia is kilogram-meters squared (kg.m^{2}).

Moment of Inertia is an important concept in Mechanics and is widely used in various fields such as structural engineering, machine design, and robotics. It is used to calculate the stability and rigidity of various structures, the efficiency of mechanical systems, and the power requirements of rotational motion.

**Recall different methods of Moment of Inertia**

Moment of Inertia is a measure of an object’s resistance to rotational motion about a particular axis. It is defined as the sum of the products of the mass of each particle in a body and its square distance from a reference axis. Moment of Inertia is a scalar quantity and has units of kilogram-meters squared (kg·m²).

There are different methods for calculating the Moment of Inertia of an object, including:

- Perpendicular Axes Theorem: This theorem states that the Moment of Inertia about any axis perpendicular to a given axis is equal to the sum of the Moments of Inertia about the two axes.
- Parallel Axes Theorem: This theorem states that the Moment of Inertia about a parallel axis can be found by subtracting the product of the object’s mass and the square of the distance between the two axes from the Moment of Inertia about the original axis.
- Composite Bodies: The Moment of Inertia of a composite body can be found by adding up the Moments of Inertia of its individual parts.
- Theorem of Perpendiculars: This theorem states that the Moment of Inertia of a body about an axis perpendicular to a plane of symmetry is equal to the sum of the Moments of Inertia about two axes that lie in that plane and are perpendicular to each other.
- Polar Moment of Inertia: This is a scalar value that represents an object’s resistance to torsion, or twisting, about a particular axis.

It’s important to note that the Moment of Inertia is dependent on the axis about which the calculation is performed, so it’s necessary to choose the appropriate axis for a given problem.

**Describe the Perpendicular-axis theorem**

The Perpendicular-axis theorem, also known as the perpendicular-axis theorem, states that the moment of inertia of a planar lamina (a flat and thin material) about an axis perpendicular to its plane is equal to the sum of the moments of inertia about any two mutually perpendicular axes lying in the plane and passing through the point of intersection of the given axis and the plane. In other words, the moment of inertia of a lamina about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two axes lying in the plane and perpendicular to each other.

The theorem is useful for finding the moment of inertia of complex objects such as beams, plates, and other structures. By breaking down the object into simpler components and calculating the moment of inertia about each of these components, the overall moment of inertia can be determined. This information is important in the design and analysis of structures and is used to determine their stability, deflection, and ability to withstand loads.

**Describe the Parallel-axis theorem**

The parallel-axis theorem, also known as the parallel-axis transfer theorem, is a fundamental theorem in engineering mechanics that relates the moment of inertia of a rigid body to its moment of inertia about a parallel axis. The theorem states that if the axis of rotation is changed from one location to another, the moment of inertia will change. The change in moment of inertia is proportional to the square of the perpendicular distance between the original and the new axis of rotation.

The theorem is expressed mathematically as: I = I_{0} + mx d^{2}, where I is the moment of inertia about the new axis, I_{0} is the moment of inertia about the original axis, m is the mass of the body, and d is the perpendicular distance between the two axes.

The parallel-axis theorem is important in engineering mechanics because it allows us to calculate the moment of inertia of a body about any axis of rotation, given the moment of inertia about a known axis. This theorem is particularly useful in the design and analysis of structures and machines, where the axis of rotation may change during operation, and in the calculation of the dynamic response of systems.

**Describe and calculate Moment of Inertia of the following sections i. Rectangular section ii. Hollow rectangular section iii. Circular section iv. Hollow circular section v. Semi-circular section**

Moment of Inertia (MOI) is a scalar property of a cross-sectional shape that represents the distribution of mass in a body and the resistance of the body to rotational motion about a given axis. It is a measure of the body’s ability to resist rotational motion and is calculated as the sum of the products of the masses of all the particles in the body and the square of their distances from a given axis of rotation.

In engineering mechanics, there are different methods to calculate the MOI of different cross-sectional shapes such as rectangles, circles, hollow shapes, and semi-circular sections.

- Rectangular section: The MOI of a rectangular section can be calculated using the formula: I = (bh
^{3})/12, where b is the width of the rectangle and h is the height of the rectangle. - Hollow rectangular section: The MOI of a hollow rectangular section can be calculated by subtracting the MOI of two smaller rectangles from the MOI of a larger rectangle that encompasses the hollow section.
- Circular section: The MOI of a circular section can be calculated using the formula: I = (π/4)d
^{4}, where d is the diameter of the circle. - Hollow circular section: The MOI of a hollow circular section can be calculated by subtracting the MOI of two smaller circles from the MOI of a larger circle that encompasses the hollow section.
- Semi-circular section: The MOI of a semi-circular section can be calculated using the formula: I = (π/32)d
^{4}, where d is the diameter of the semi-circular section.

It is important to note that the MOI of a cross-sectional shape depends on the axis of rotation and the orientation of the shape. Hence, different methods may be required for different shapes and orientations.

**Describe and calculate Moment of Inertia of the following sections i. Triangular section ii. Cut-out section iii. Composite section iv. Built-up section**

The term “Moment of Inertia” refers to a property of a body that determines the distribution of its mass relative to a given axis. This property affects the way a body rotates about that axis and is related to the body’s resistance to rotational motion. Moment of inertia is calculated for different shapes and sections to study their behavior under rotation.

To calculate the Moment of Inertia of different sections, various methods are used, including mathematical formulas and physical experiments. The methods used to calculate Moment of Inertia depend on the shape of the section being studied.

Here are the calculations for Moment of Inertia of various sections:

i. Triangular Section: The Moment of Inertia of a triangular section can be calculated using the formula, I = bh^{3}/36, where b is the base and h is the height of the triangle.

ii. Cut-out Section: The Moment of Inertia of a cut-out section is calculated by subtracting the Moment of Inertia of the removed part from the original section.

iii. Composite Section: The Moment of Inertia of a composite section is calculated by summing the Moments of Inertia of its constituent parts.

iv. Built-up Section: The Moment of Inertia of a built-up section is calculated by summing the Moments of Inertia of its individual parts and considering the distribution of its mass relative to the axis of rotation.