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Questions and Answers
What characterizes the motion of a rigid body in terms of its deformation?
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?
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?
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?
When a rigid body does not have translational motion, what type of motion does it exhibit?
How do real bodies behave under the influence of forces?
How do real bodies behave under the influence of forces?
Which of the following describes translational motion in the context of a rigid body?
Which of the following describes translational motion in the context of a rigid body?
If a rectangular block slides down an inclined plane, how is it treated in terms of rigid body motion?
If a rectangular block slides down an inclined plane, how is it treated in terms of rigid body motion?
Which of the following scenarios can be considered a situation where rigid body assumptions are valid?
Which of the following scenarios can be considered a situation where rigid body assumptions are valid?
What is the relationship between the centre of mass and geometric centre for a homogeneous thin rod?
What is the relationship between the centre of mass and geometric centre for a homogeneous thin rod?
How can the centre of mass for homogeneous bodies like rings or spheres be described categorically?
How can the centre of mass for homogeneous bodies like rings or spheres be described categorically?
What does the integral equation ( R = \frac{1}{M} \int r \ dm ) represent?
What does the integral equation ( R = \frac{1}{M} \int r \ dm ) represent?
In the equation for the centre of mass, what does the variable ( M ) represent?
In the equation for the centre of mass, what does the variable ( M ) represent?
What is the main reason the centre of mass of the L-shaped object is not at the geometric centre of triangle OAB?
What is the main reason the centre of mass of the L-shaped object is not at the geometric centre of triangle OAB?
Which method can be used to find the centre of mass for a triangular lamina?
Which method can be used to find the centre of mass for a triangular lamina?
What condition allows the centre of mass to lie on the median LP in the triangular lamina?
What condition allows the centre of mass to lie on the median LP in the triangular lamina?
What is implied by the centre of mass lying on the line OD for the L-shaped lamina?
What is implied by the centre of mass lying on the line OD for the L-shaped lamina?
If the three squares of the L-shaped lamina had different masses, how would that affect the centre of mass?
If the three squares of the L-shaped lamina had different masses, how would that affect the centre of mass?
What total mass is utilized when calculating the Y-coordinate of the centre of mass for a triangular lamina?
What total mass is utilized when calculating the Y-coordinate of the centre of mass for a triangular lamina?
What would change in the method if the masses of the squares making up the L-shaped lamina were not equal?
What would change in the method if the masses of the squares making up the L-shaped lamina were not equal?
How is the centre of mass of a system of particles generally determined?
How is the centre of mass of a system of particles generally determined?
What is the expression for the force acting on the second particle in a system of n particles?
What is the expression for the force acting on the second particle in a system of n particles?
Which statement best describes how external forces are treated in a system of particles?
Which statement best describes how external forces are treated in a system of particles?
What is the linear momentum of the entire system of n particles defined as?
What is the linear momentum of the entire system of n particles defined as?
Why was the analysis of rotational and internal motions previously omitted in earlier studies?
Why was the analysis of rotational and internal motions previously omitted in earlier studies?
In the context of a system with multiple particles, what denotes the mass of the first particle?
In the context of a system with multiple particles, what denotes the mass of the first particle?
What mathematical form represents the total force acting on a system of n particles?
What mathematical form represents the total force acting on a system of n particles?
What does the equation P = p_1 + p_2 + ... + p_n represent in the system of n particles?
What does the equation P = p_1 + p_2 + ... + p_n represent in the system of n particles?
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?
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?
What describes the motion of each particle in a rigid body during rotation about a fixed axis?
What describes the motion of each particle in a rigid body during rotation about a fixed axis?
Which of the following statements is true regarding the radius of the circular motion of a particle in rotation?
Which of the following statements is true regarding the radius of the circular motion of a particle in rotation?
In the context of rigid body rotation, what happens to a particle located directly on the axis of rotation?
In the context of rigid body rotation, what happens to a particle located directly on the axis of rotation?
How does the motion of the blades of a fan differ from the motion of the pivot of the fan?
How does the motion of the blades of a fan differ from the motion of the pivot of the fan?
What type of motion is described by a top spinning in place?
What type of motion is described by a top spinning in place?
In the diagram of rigid body rotation, what is the significance of points P1, P2, and P3?
In the diagram of rigid body rotation, what is the significance of points P1, P2, and P3?
What characteristic defines the plane in which the circular motion occurs during the rotation of a rigid body?
What characteristic defines the plane in which the circular motion occurs during the rotation of a rigid body?
Which term best describes the relationship between the center of circular motion and the fixed axis in rigid body rotation?
Which term best describes the relationship between the center of circular motion and the fixed axis in rigid body rotation?
What is the primary assumption made when treating a rigid body as a continuous distribution of mass?
What is the primary assumption made when treating a rigid body as a continuous distribution of mass?
In the context of the center of mass for a thin rod, which axis is typically aligned with the length of the rod?
In the context of the center of mass for a thin rod, which axis is typically aligned with the length of the rod?
How does increasing the number of elements (n) and decreasing their mass (∆mi) affect the computation of the center of mass?
How does increasing the number of elements (n) and decreasing their mass (∆mi) affect the computation of the center of mass?
What symmetry is assumed when calculating the center of mass for a thin rod centered at the origin?
What symmetry is assumed when calculating the center of mass for a thin rod centered at the origin?
What represents the cumulative mass in the integral notation for the total mass of the body?
What represents the cumulative mass in the integral notation for the total mass of the body?
What condition must be true for the integral to become zero regarding the pairs of mass elements in a symmetrical rod?
What condition must be true for the integral to become zero regarding the pairs of mass elements in a symmetrical rod?
What is the correct expression for finding the coordinates of the center of mass for a system of particles?
What is the correct expression for finding the coordinates of the center of mass for a system of particles?
When treating the mass of a rigid body as continuous, how are the individual mass elements denoted?
When treating the mass of a rigid body as continuous, how are the individual mass elements denoted?
What geometrical feature does the centre of mass of a homogeneous thin rod coincide with?
What geometrical feature does the centre of mass of a homogeneous thin rod coincide with?
Which integral expression represents the coordinates of the centre of mass for a three-dimensional body?
Which integral expression represents the coordinates of the centre of mass for a three-dimensional body?
What does the total mass, represented by M, signify in the context of calculating centre of mass?
What does the total mass, represented by M, signify in the context of calculating centre of mass?
How does reflection symmetry apply to the centre of mass of homogeneous bodies?
How does reflection symmetry apply to the centre of mass of homogeneous bodies?
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?
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?
Which of the following shapes does NOT have the center of mass coincide with the geometric center when it is homogeneous?
Which of the following shapes does NOT have the center of mass coincide with the geometric center when it is homogeneous?
What does the integral $\int y \ dm$ help to determine in the context of centre of mass?
What does the integral $\int y \ dm$ help to determine in the context of centre of mass?
Which statement correctly describes the centre of mass for thick rods with rectangular cross sections?
Which statement correctly describes the centre of mass for thick rods with rectangular cross sections?
Flashcards
Rigid Body
Rigid Body
A body that does not deform significantly under applied forces.
Translational Motion
Translational Motion
The motion of a body in which all points move in parallel paths.
Rotational Motion
Rotational Motion
The motion of a rigid body that revolves around a fixed axis.
Axis of Rotation
Axis of Rotation
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Deformation
Deformation
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Negligible Deformation
Negligible Deformation
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Constraining a Rigid Body
Constraining a Rigid Body
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Motion Examples
Motion Examples
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Center of Mass
Center of Mass
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Coordinates of Center of Mass
Coordinates of Center of Mass
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Homogenous Thin Rod
Homogenous Thin Rod
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Geometric center
Geometric center
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Reflection Symmetry
Reflection Symmetry
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Vector Expression for Center of Mass (R)
Vector Expression for Center of Mass (R)
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Center of Mass and Reflection symmetry relationship
Center of Mass and Reflection symmetry relationship
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Example 6.1
Example 6.1
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Axis of Rotation
Axis of Rotation
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Rotational Motion
Rotational Motion
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Rigid Body
Rigid Body
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Particle's Path
Particle's Path
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Radius of Circle
Radius of Circle
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Fixed Axis
Fixed Axis
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Rotating Blades
Rotating Blades
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Oscillating Axis
Oscillating Axis
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Center of Mass (X, Y, Z)
Center of Mass (X, Y, Z)
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Continuous Mass Distribution
Continuous Mass Distribution
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Homogenous Thin Rod
Homogenous Thin Rod
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Reflection Symmetry
Reflection Symmetry
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Geometric Centre
Geometric Centre
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Center of Mass of Thin Rod
Center of Mass of Thin Rod
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Center of Mass and Reflection Symmetry
Center of Mass and Reflection Symmetry
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Integral Method for Center of Mass calculation
Integral Method for Center of Mass calculation
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Centre of Mass (L-shape)
Centre of Mass (L-shape)
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Centre of Mass (triangle)
Centre of Mass (triangle)
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Centre of Mass and Symmetry
Centre of Mass and Symmetry
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Geometric Centre vs. Centre of Mass
Geometric Centre vs. Centre of Mass
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Variable Mass (L-shape)
Variable Mass (L-shape)
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Calculating Centre of Mass
Calculating Centre of Mass
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L-shape lamina
L-shape lamina
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Lamina
Lamina
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Center of Mass
Center of Mass
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Linear Momentum
Linear Momentum
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System of Particles
System of Particles
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External Forces
External Forces
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Translational Motion
Translational Motion
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Linear Momentum of System
Linear Momentum of System
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System of n Particles
System of n Particles
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Eq. (6.10)
Eq. (6.10)
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Study Notes
Systems of Particles and Rotational Motion
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Introduction: Early chapters focused on individual particle motion. This chapter expands to encompass extended bodies (finite size), crucial for real-world scenarios.
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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.
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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 $\textbf{F}{ext} = M\textbf{a}{CM}$.
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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.
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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.
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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.
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Angular acceleration: The time rate of change of angular velocity is angular acceleration (α). α = dω / dt.
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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.
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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).
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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.
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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.
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Kinematics of Rotation: Describing rotational motion without considering the cause of the motion.
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Dynamics of Rotation: Describing rotational motion and the causes (forces).
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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.
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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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