5.2 Center of gravity and mass moment of inertia

Center of gravity and mass moment of inertia are crucial concepts in understanding how objects behave under forces. They help us figure out where an object's weight acts and how it resists rotational motion.

These ideas are key in designing stable structures and efficient machines. By grasping these concepts, we can predict how objects will balance, rotate, and respond to applied forces in real-world situations.

Centroids vs Centers of Gravity

Geometric Center and Mass Distribution

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• Centroids are the geometric center of a shape or object, calculated solely based on the geometry of the object
• It is the point where all the area or volume could be concentrated and still have the same first moment of area or volume about any axis
• Centers of gravity are the point where an object's total weight can be considered to act
• It factors in both the geometry and the density distribution of the object

Uniform vs Non-Uniform Density

• For objects with uniform density, the centroid and center of gravity coincide (solid cylinder)
• For objects with non-uniform density, the center of gravity depends on the distribution of mass and may not be at the geometric center (car engine)
• The center of gravity is important for understanding the stability and balance of objects (leaning tower of Pisa)
• It also determines their behavior when subjected to forces and moments (tipping of a crane)

Center of Gravity Calculation

Dividing Objects into Smaller Parts

• To calculate the center of gravity for objects with non-uniform density, the object must be divided into smaller parts or elements with known or approximated densities (sliced bread loaf)
• The center of gravity for each individual part is determined based on its geometry and density (individual slices of bread)
• The overall center of gravity is then calculated using a weighted average of the individual centers of gravity, with the weights being the masses of each part

Integration Techniques for Continuous Objects

• For continuous objects with varying density, integration techniques may be necessary to determine the center of gravity (graduated cylinder filled with liquid)
• The formula for calculating the center of gravity in a particular direction (x, y, or z) is: $X_cg = (∫ x dm) / (∫ dm)$, where dm is the differential mass element and the integrals are taken over the entire object
• This formula represents the weighted average of the position coordinates, with the mass acting as the weighting factor (density variations in a metal rod)

Mass Moment of Inertia

Resistance to Rotational Acceleration

• Mass moment of inertia is a measure of an object's resistance to rotational acceleration about a specific axis
• It depends on the distribution of mass within the object relative to the axis of rotation (figure skater spinning with arms extended vs tucked in)
• The mass moment of inertia is analogous to mass in translational motion, as it relates the applied torque to the resulting angular acceleration through the equation: $T = I * α$, where T is the torque, I is the mass moment of inertia, and α is the angular acceleration
• Objects with a larger mass moment of inertia require more torque to achieve the same angular acceleration compared to objects with a smaller mass moment of inertia (rotating a long rod vs a short rod)

Importance in Rotational Dynamics

• The mass moment of inertia is important in understanding the rotational dynamics of objects (spinning top)
• It plays a crucial role in the design of flywheels, gears, and other rotating machinery (car engine components)
• The parallel axis theorem allows for the calculation of the mass moment of inertia about any parallel axis, given the mass moment of inertia about an axis passing through the center of gravity and the perpendicular distance between the two axes (rotating a door about its hinges)

Mass Moment of Inertia for Shapes

Formulas for Simple Geometric Shapes

• The mass moment of inertia for various shapes and solids can be determined using predefined formulas or by integration techniques
• For simple geometric shapes, such as rectangular plates, circular disks, and solid cylinders, the mass moment of inertia can be calculated using the following formulas:
  • Rectangular plate (about an axis perpendicular to the plate and passing through its center): $I = (1/12) * m * (a^2 + b^2)$, where m is the mass, a is the length, and b is the width
  • Circular disk (about an axis perpendicular to the disk and passing through its center): $I = (1/2) * m * r^2$, where m is the mass and r is the radius
  • Solid cylinder (about its central axis): $I = (1/2) * m * r^2$, where m is the mass and r is the radius

Complex Shapes and Integration Techniques

• For more complex shapes or composite objects, the mass moment of inertia can be determined by dividing the object into simpler parts, calculating the mass moment of inertia for each part, and then using the parallel axis theorem to find the total mass moment of inertia about the desired axis (I-beam)
• In some cases, integration techniques may be necessary to calculate the mass moment of inertia, especially for objects with non-uniform density or irregular shapes (airplane wing)
• These techniques involve setting up and evaluating appropriate integrals based on the object's geometry and density distribution (3D-printed object with varying infill density)