Physics Chapter: Center of Gravity & Inertia

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Questions and Answers

What is the center of gravity of a body dependent on?

The surface area of the body

The shape of the body only

The weight of all particles and their positions (correct)

The density of the material only

Which equation represents the total weight of the body?

$W = \sum F_z$ (correct)

$W = m \cdot g$

$W = \int dW$ (correct)

$W = V \cdot d$

How can the location of the center of gravity be determined with respect to the y-axis?

By equating the moments of the weights about the y-axis (correct)

By calculating the force exerted at the top of the y-axis

By averaging the distances of all particles from the y-axis

By using the density of the material along the y-axis

What must be true for the three-dimensional location of mass center G?

Moments about all three axes must be considered Signup and view all the answers

In the context of mass moment of inertia, what does the mass moment represent?

The rotational inertia about an axis Signup and view all the answers

Which statement is true regarding the center of mass of a wire compared to that of a plate?

The center of mass is influenced by the shape and distribution of mass Signup and view all the answers

What is required to calculate the center of mass in a two-dimensional body?

Moments about both axes must be determined Signup and view all the answers

What is the significance of the density in the calculation of center of gravity?

It affects the forces acting on different parts of the body Signup and view all the answers

What does the product $rac{ riangle m r^2}{ riangle m}$ represent in rotational motion?

Moment of inertia with respect to axis AA’ Signup and view all the answers

How is the radius of gyration $k$ defined with respect to the moment of inertia $I$?

$k = rac{I}{m}$ Signup and view all the answers

In the context of angular motion, which of the following equations applies to calculate the tangential force acting on a particle?

$F_t = m a_t$ Signup and view all the answers

What is the effect of increasing the number of elements when calculating the moment of inertia?

It increases the accuracy of the integral limit Signup and view all the answers

Which of the following is true regarding the moment of inertia with respect to coordinate axes?

It is dependent on the distribution of mass around the axis Signup and view all the answers

In SI units, what is the unit of mass moment of inertia?

kg.m^2 Signup and view all the answers

What does the equation $I = riangle m r^2$ signify?

The measure of resistance to rotation Signup and view all the answers

Which statement about mass moment of inertia is incorrect?

It depends only on the mass of the object. Signup and view all the answers

Study Notes

Center of Gravity

A body is composed of countless tiny particles.
Each particle within a gravitational field has a weight (dW).
These weights form a parallel force system.
The resultant of this system is the total weight of the body.
The resultant force passes through a single point called the center of gravity (G*).
The total weight (W) of a body is the sum of the weights of all its particles (∑ Fz).
W = ∫dW
The center of gravity's position (x, y, z) is determined by summing moments about each axis.
x = (∫xdW)/W
y = (∫ydW)/W
z = (∫zdW)/W

Mass Moment of Inertia

Consider a small mass (Am) on a rod (AA') that rotates freely.
If a couple (M) due to a tangent force (F) is applied, the system rotates.
Applying Newton's 2nd law in the tangential direction leads to: ∑ Ft = m*a_t.
Multiplying by radius (r) gives: Fr = rΔma_t.
M = Amr²α
This product (Amr²) measures rotational inertia (the resistance to changes in rotation).
Amr² is the mass moment of inertia of Am relative to AA'.
For a larger body, divide it into many small elements (Am₁, Am₂, etc.).
The body's total resistance to rotation is the sum of the moments of inertia of these elements.
I = ∫r²dm

Radius of Gyration

The radius of gyration (k) shows the distance at which the full mass would need to be concentrated for the moment of inertia to remain the same.
I = mk²
k = √(I/m)
SI units: k is in meters, m is in kilograms, and I is in kg⋅m².

Parallel Axes Theorem

To calculate moment of inertia about a different axis (parallel to the original axis), use the theorem.
Ix = Ix' + md_x²
Iy = Iy' + md_y²
Iz = Iz' + md_z²
where (x, y, z) are the new coordinates, and (x', y', z') are the original coordinates and d_x, d_y, d_z are the distances between the parallel axes.

Examples of Mass Moments of Inertia

Presented are formulas for the mass moments of inertia for various shapes (slender rod, thin rectangular plate, rectangular prism, thin disk, circular cylinder, circular cone, sphere). These formulas are related to the shape's dimensions and mass.

Example Problem (Page 12)

The example provides a pendulum system with a description of different components.
The problem asks to find the center of mass, and the moment of inertia about specific axes.

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Description

This quiz explores the concepts of center of gravity and mass moment of inertia. It covers how gravitational forces act on particles within a body and how these concepts relate to rotational motion. Test your knowledge on the calculations and principles involved in these fundamental physics topics.