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?

    <p>Moments about all three axes must be considered (C)</p> Signup and view all the answers

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

    <p>The rotational inertia about an axis (C)</p> Signup and view all the answers

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

    <p>The center of mass is influenced by the shape and distribution of mass (C)</p> Signup and view all the answers

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

    <p>Moments about both axes must be determined (A)</p> Signup and view all the answers

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

    <p>It affects the forces acting on different parts of the body (A)</p> Signup and view all the answers

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

    <p>Moment of inertia with respect to axis AA’ (B)</p> Signup and view all the answers

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

    <p>$k = rac{I}{m}$ (C)</p> 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?

    <p>$F_t = m a_t$ (D)</p> Signup and view all the answers

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

    <p>It increases the accuracy of the integral limit (C)</p> Signup and view all the answers

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

    <p>It is dependent on the distribution of mass around the axis (B)</p> Signup and view all the answers

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

    <p>kg.m^2 (D)</p> Signup and view all the answers

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

    <p>The measure of resistance to rotation (A)</p> Signup and view all the answers

    Which statement about mass moment of inertia is incorrect?

    <p>It depends only on the mass of the object. (C)</p> Signup and view all the answers

    Flashcards

    What is the center of gravity?

    The point where the entire weight of a body can be considered to act, representing the balance point.

    How do you find the center of gravity?

    The location of the center of gravity is determined by equating the moment of the total weight (W) about an axis to the sum of moments of individual weight elements about the same axis

    What is the mass moment of inertia?

    A scalar that represents a body's resistance to rotational motion around a specific axis, measured in kg*m^2

    What factors influence mass moment of inertia?

    It depends on the mass distribution relative to the axis of rotation: further mass from the axis means more inertia, thus harder to rotate.

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    What is the significance of mass moment of inertia?

    It represents the tendency of a body to resist changes in its rotational motion.

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    How do you calculate the mass moment of inertia?

    The mass moment of inertia is calculated by summing the product of each mass element and the square of its distance from the axis of rotation.

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    What are the applications of mass moment of inertia?

    It helps determine the rotational kinetic energy of an object and the forces required to change its angular velocity.

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    Is mass moment of inertia constant for a given object?

    The mass moment of inertia depends on the shape and mass distribution of the object, as well as the chosen axis of rotation.

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    Moment of Inertia

    The resistance of a body to changes in its rotational motion. It is a measure of how difficult it is to start or stop an object from rotating.

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    Tangent Force

    A force that acts tangentially to a circular path, causing an object to rotate.

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    Radius of Rotation

    The distance from the axis of rotation to a point on the object.

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    Newton's 2nd Law of Rotational Motion

    The angular acceleration of a rigid body is directly proportional to the net torque acting on the body and inversely proportional to the body's moment of inertia.

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    Mass Moment of Inertia of a Particle

    The product of a particle's mass and the square of its distance from the axis of rotation.

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    Mass Moment of Inertia of a Body

    The sum of the mass moments of inertia of all the particles in a rigid body. It quantifies the body's resistance to changes in its rotational motion.

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    Radius of Gyration

    The distance from the axis of rotation to a point where the entire mass of the body can be concentrated without changing its moment of inertia.

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    Moment of Inertia Formula

    The integral of the product of the square of the distance from the axis of rotation and the mass element of the body.

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    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*at.
    • Multiplying by radius (r) gives: Fr = rΔmat.
    • 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' + mdx²
    • Iy = Iy' + mdy²
    • Iz = Iz' + mdz²
    • where (x, y, z) are the new coordinates, and (x', y', z') are the original coordinates and dx, dy, dz 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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    Lecture 12 Mass Moment of Inertia PDF

    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.

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