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What characterizes the motion of a rigid body in terms of its deformation?

  • It only deforms under extreme conditions.
  • It remains completely unchanged regardless of external forces.
  • It may deform, but such deformations are often negligible. (correct)
  • It deforms permanently under all forces.

    • Which of the following is true for the axis of rotation in rigid body motion?

  • It is always positioned externally to the body.
  • It defines the line about which the body rotates. (correct)
  • It is the same as the center of mass.
  • It can change position during motion.

    • What is a common example of rotational motion of a rigid body?

  • A marble rolling across the floor.
  • A car moving in a straight line.
  • A block sliding down an inclined plane.
  • A ceiling fan spinning around its central shaft. (correct)

    • When a rigid body does not have translational motion, what type of motion does it exhibit?

    <p>Rotational motion.</p> Signup and view all the answers

    How do real bodies behave under the influence of forces?

    <p>They may warp or bend.</p> Signup and view all the answers

    Which of the following describes translational motion in the context of a rigid body?

    <p>Movement where every particle of the body moves together.</p> Signup and view all the answers

    If a rectangular block slides down an inclined plane, how is it treated in terms of rigid body motion?

    <p>As a rigid body where all particles move together.</p> Signup and view all the answers

    Which of the following scenarios can be considered a situation where rigid body assumptions are valid?

    <p>The spinning of a planet around its axis.</p> Signup and view all the answers

    What is the relationship between the centre of mass and geometric centre for a homogeneous thin rod?

    <p>The centre of mass is located at the geometric centre.</p> Signup and view all the answers

    How can the centre of mass for homogeneous bodies like rings or spheres be described categorically?

    <p>Coincides with their geometric centre.</p> Signup and view all the answers

    What does the integral equation ( R = \frac{1}{M} \int r \ dm ) represent?

    <p>The centre of mass of the body.</p> Signup and view all the answers

    In the equation for the centre of mass, what does the variable ( M ) represent?

    <p>Total mass of the body.</p> Signup and view all the answers

    What is the main reason the centre of mass of the L-shaped object is not at the geometric centre of triangle OAB?

    <p>The masses of the squares are distributed unevenly.</p> Signup and view all the answers

    Which method can be used to find the centre of mass for a triangular lamina?

    <p>By subdividing it into narrow strips parallel to one side.</p> Signup and view all the answers

    What condition allows the centre of mass to lie on the median LP in the triangular lamina?

    <p>Each strip has its centre of mass at its midpoint.</p> Signup and view all the answers

    What is implied by the centre of mass lying on the line OD for the L-shaped lamina?

    <p>The arrangement of mass affects balance.</p> Signup and view all the answers

    If the three squares of the L-shaped lamina had different masses, how would that affect the centre of mass?

    <p>It could shift towards the heavier square.</p> Signup and view all the answers

    What total mass is utilized when calculating the Y-coordinate of the centre of mass for a triangular lamina?

    <p>The sum of the mass of all strips.</p> Signup and view all the answers

    What would change in the method if the masses of the squares making up the L-shaped lamina were not equal?

    <p>Weights must be considered in the calculations.</p> Signup and view all the answers

    How is the centre of mass of a system of particles generally determined?

    <p>By averaging the positions based on mass.</p> Signup and view all the answers

    What is the expression for the force acting on the second particle in a system of n particles?

    <p>F = m_2 a_2</p> Signup and view all the answers

    Which statement best describes how external forces are treated in a system of particles?

    <p>External forces are assumed to act at the system's center of mass.</p> Signup and view all the answers

    What is the linear momentum of the entire system of n particles defined as?

    <p>The vector sum of the momenta of individual particles.</p> Signup and view all the answers

    Why was the analysis of rotational and internal motions previously omitted in earlier studies?

    <p>The effects of these motions were assumed to be unimportant.</p> Signup and view all the answers

    In the context of a system with multiple particles, what denotes the mass of the first particle?

    <p>m_1</p> Signup and view all the answers

    What mathematical form represents the total force acting on a system of n particles?

    <p>F = F_1 + F_2 + ... + F_n</p> Signup and view all the answers

    What does the equation P = p_1 + p_2 + ... + p_n represent in the system of n particles?

    <p>The total linear momentum of the system.</p> Signup and view all the answers

    What does it mean for the forces acting on the system when it says they can be assumed to act at the center of mass?

    <p>The resultant motion can be simplified to a single point.</p> Signup and view all the answers

    What describes the motion of each particle in a rigid body during rotation about a fixed axis?

    <p>The particles move in circles with their centers on the axis.</p> Signup and view all the answers

    Which of the following statements is true regarding the radius of the circular motion of a particle in rotation?

    <p>The radius is determined by the distance from the particle to the axis.</p> Signup and view all the answers

    In the context of rigid body rotation, what happens to a particle located directly on the axis of rotation?

    <p>It remains stationary.</p> Signup and view all the answers

    How does the motion of the blades of a fan differ from the motion of the pivot of the fan?

    <p>The pivot does not move, while the blades rotate around it.</p> Signup and view all the answers

    What type of motion is described by a top spinning in place?

    <p>Rotation about a non-fixed axis.</p> Signup and view all the answers

    In the diagram of rigid body rotation, what is the significance of points P1, P2, and P3?

    <p>They represent particles at varying distances from the axis.</p> Signup and view all the answers

    What characteristic defines the plane in which the circular motion occurs during the rotation of a rigid body?

    <p>It is perpendicular to the axis of rotation.</p> Signup and view all the answers

    Which term best describes the relationship between the center of circular motion and the fixed axis in rigid body rotation?

    <p>The center of circular motion lies perpendicular to the axis.</p> Signup and view all the answers

    What is the primary assumption made when treating a rigid body as a continuous distribution of mass?

    <p>The spacing of the particles is small.</p> Signup and view all the answers

    In the context of the center of mass for a thin rod, which axis is typically aligned with the length of the rod?

    <p>X-axis</p> Signup and view all the answers

    How does increasing the number of elements (n) and decreasing their mass (∆mi) affect the computation of the center of mass?

    <p>It allows for an exact representation using integrals.</p> Signup and view all the answers

    What symmetry is assumed when calculating the center of mass for a thin rod centered at the origin?

    <p>Reflection symmetry</p> Signup and view all the answers

    What represents the cumulative mass in the integral notation for the total mass of the body?

    <p>∫dm</p> Signup and view all the answers

    What condition must be true for the integral to become zero regarding the pairs of mass elements in a symmetrical rod?

    <p>For every dm at x, there must be another dm at –x.</p> Signup and view all the answers

    What is the correct expression for finding the coordinates of the center of mass for a system of particles?

    <p>X = ∑(m_i * x_i) / ∑m_i</p> Signup and view all the answers

    When treating the mass of a rigid body as continuous, how are the individual mass elements denoted?

    <p>∆m1, ∆m2, ∆mn</p> Signup and view all the answers

    What geometrical feature does the centre of mass of a homogeneous thin rod coincide with?

    <p>The geometric centre</p> Signup and view all the answers

    Which integral expression represents the coordinates of the centre of mass for a three-dimensional body?

    <p>$X = \frac{1}{M} \int x \ dm$</p> Signup and view all the answers

    What does the total mass, represented by M, signify in the context of calculating centre of mass?

    <p>The entirety of mass distribution in the body</p> Signup and view all the answers

    How does reflection symmetry apply to the centre of mass of homogeneous bodies?

    <p>It allows equal distribution of elements about the origin.</p> Signup and view all the answers

    In the context of the given content, what would be the centre of mass if it is chosen as the origin in a coordinate system?

    <p>0 if the system is symmetrical</p> Signup and view all the answers

    Which of the following shapes does NOT have the center of mass coincide with the geometric center when it is homogeneous?

    <p>An asymmetric irregular shape</p> Signup and view all the answers

    What does the integral $\int y \ dm$ help to determine in the context of centre of mass?

    <p>The y-coordinate of the centre of mass</p> Signup and view all the answers

    Which statement correctly describes the centre of mass for thick rods with rectangular cross sections?

    <p>It coincides with the geometric centre due to symmetry.</p> Signup and view all the answers

    Study Notes

    Systems of Particles and Rotational Motion

    • Introduction: Early chapters focused on individual particle motion. This chapter expands to encompass extended bodies (finite size), crucial for real-world scenarios.

    • Centre of mass: The centre of mass (CM) of a system is the weighted average position of the constituent particles. It is vital for understanding the motion of extended bodies. For a system of two particles, the CM is located at a point along the straight line joining the particles. Its location for multiple particles is defined by a weighted average formula.

    • Motion of the center of mass: The motion/movement of the CM of a system of particles is determined by the external forces on the system, ignoring internal forces. The external force acting on the particles and the acceleration of the CM of the system are related by $\textbf{F}{ext} = M\textbf{a}{CM}$.

    • Rigid bodies: A rigid body maintains a constant shape under the influence of forces, an idealization. Real bodies deform but are treated as rigid in many situations like wheels or beams for simplification purposes.

    • Types of rigid body motion: Rigid body motion encompasses translational and rotational movements.

      • Pure translation: All particles of the body move with the same velocity at any given moment.
      • Pure rotation: All particles move in circles with the same angular velocity about a fixed axis.
      • Combination of translation and rotation, which is a combination of the above two types.

    • Angular velocity: The angular velocity (ω) describes rotational motion. The magnitude of the linear velocity (v) is related to it by v = ωr. The vector @ is directed along the axis of rotation.

    • Angular acceleration: The time rate of change of angular velocity is angular acceleration (α). α = dω / dt.

    • Torque: A vector quantity representing the rotational effect of a force (analogous to linear force). Torque (τ) is the product of force (F) and the perpendicular distance from the axis of rotation to the line of action of the force: τ = r x F where r is the position vector of point where the force is applied with respect to the axis of rotation. In vector notation, τ = rF sinθ, or τ = r F⊥ where r⊥ is the component of r perpendicular to F.

    • Conservation of Angular Momentum: When there is no net external torque on a system, its total angular momentum remains constant. This principle is analogous to conservation of linear momentum (when the net force is zero).

    • Moment of Inertia: A measure of how difficult it is for a body to change its rotational motion about an axis. It is a body-specific property that depends on the mass distribution within a body: I = Σ mr². The larger the moment of inertia, the more difficult it is to start, stop, or change the speed of rotation of the body.

    • Center of Mass and Center of Gravity (CG): The centre of mass (CM) is the average position of all the masses in a body, while the centre of gravity (CG) is the point where the resultant gravitational force from all parts on a body acts. For uniform gravitational fields, the CM and CG coincide.

    • Kinematics of Rotation: Describing rotational motion without considering the cause of the motion.

    • Dynamics of Rotation: Describing rotational motion and the causes (forces).

    • Equilibrium of a Rigid Body: A rigid body is in equilibrium if there's no net force and no net torque acting on it. The vector sum of all forces is zero. The sum of all torques is zero.

    • Principle of Moments: The principle of moments states that for a body in rotational equilibrium, the sum of clockwise moments is equal to the sum of anticlockwise moments around a fixed point.

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