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What determines the final velocity of a rocket?
What determines the final velocity of a rocket?
In an oblique collision, which component of momentum is conserved?
In an oblique collision, which component of momentum is conserved?
What characterizes a perfectly inelastic collision?
What characterizes a perfectly inelastic collision?
What is the coefficient of restitution for perfectly elastic collisions?
What is the coefficient of restitution for perfectly elastic collisions?
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What happens to kinetic energy in a perfectly inelastic collision?
What happens to kinetic energy in a perfectly inelastic collision?
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What is the primary focus of kinematics?
What is the primary focus of kinematics?
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Which principle is the impulse and momentum method developed from?
Which principle is the impulse and momentum method developed from?
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What defines an internal force within a system of particles?
What defines an internal force within a system of particles?
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How does the center of mass behave when a force is applied at it?
How does the center of mass behave when a force is applied at it?
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What is an example of an extended body?
What is an example of an extended body?
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Which of the following accurately describes a rigid body?
Which of the following accurately describes a rigid body?
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What characterizes the center of mass of a system of discrete particles?
What characterizes the center of mass of a system of discrete particles?
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Which statement is true about external forces acting on a system of particles?
Which statement is true about external forces acting on a system of particles?
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What is linear impulse defined as?
What is linear impulse defined as?
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How is linear momentum calculated?
How is linear momentum calculated?
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What does the Impulse-Momentum Principle state?
What does the Impulse-Momentum Principle state?
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What characterizes impulsive motion?
What characterizes impulsive motion?
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What is the role of non-impulsive forces during impulsive motion?
What is the role of non-impulsive forces during impulsive motion?
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How is the velocity of the center of mass calculated for a system of two bodies?
How is the velocity of the center of mass calculated for a system of two bodies?
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In vector form, how is the velocity of the center of mass determined for a system of particles?
In vector form, how is the velocity of the center of mass determined for a system of particles?
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What is the condition for the conservation of linear momentum in a system of particles?
What is the condition for the conservation of linear momentum in a system of particles?
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What is the relationship between the center of mass (CM) of a system and the net external force acting on it?
What is the relationship between the center of mass (CM) of a system and the net external force acting on it?
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Which statement about Newton's third law in the context of systems of particles is correct?
Which statement about Newton's third law in the context of systems of particles is correct?
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How can the acceleration of the center of mass in a system of two blocks connected by an unstretched spring be determined?
How can the acceleration of the center of mass in a system of two blocks connected by an unstretched spring be determined?
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What happens to the total mechanical energy of a system when no external forces act and the internal forces do no work?
What happens to the total mechanical energy of a system when no external forces act and the internal forces do no work?
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Which of the following correctly defines impulsive forces?
Which of the following correctly defines impulsive forces?
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What defines a variable mass system?
What defines a variable mass system?
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To calculate the work done by internal forces in a system, which concept is crucial?
To calculate the work done by internal forces in a system, which concept is crucial?
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What does the principle of conservation of momentum imply for a system with no external impulsive forces?
What does the principle of conservation of momentum imply for a system with no external impulsive forces?
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How is the center of mass calculated for a system of discrete particles?
How is the center of mass calculated for a system of discrete particles?
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What does the center of mass of a two-particle system depend on?
What does the center of mass of a two-particle system depend on?
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In the Cartesian coordinates, how is the x-coordinate of the center of mass expressed?
In the Cartesian coordinates, how is the x-coordinate of the center of mass expressed?
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What is the equation for the center of mass (CM) of an extended body?
What is the equation for the center of mass (CM) of an extended body?
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How does the motion of a system of particles relate to the center of mass?
How does the motion of a system of particles relate to the center of mass?
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What is true about the centroidal frame of reference?
What is true about the centroidal frame of reference?
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Where does the center of mass of a uniform symmetrical body lie?
Where does the center of mass of a uniform symmetrical body lie?
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What happens to the center of mass when two equal masses are separated by a distance?
What happens to the center of mass when two equal masses are separated by a distance?
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Study Notes
Study of Kinematics vs. Kinetics
- Kinematics studies motion without considering forces and energy.
- Kinetics investigates the effects of forces and energy on motion.
- Kinetics incorporates Newton's Laws of motion, work-energy methods, and impulse and momentum methods.
- Work-energy and impulse-momentum methods are derived from Newton's second law (F = ma).
- These methods simplify analysis by avoiding direct calculation of acceleration.
- Work and energy methods relate force, mass, velocity, and displacement.
- Impulse and momentum methods connect force, mass, and the duration of force application.
System of Particles
- A system of particles is a collection of interacting or non-interacting particles, including actual particles or parts of rigid bodies.
- Internal forces act between particles within a system.
- External forces act on particles from sources outside the system.
- Extended bodies are systems with infinitely many particles, with infinitely small distances between them.
- Deformable bodies change the separation and relative positions of their particles.
- Rigid bodies maintain constant separation and relative positions between particles.
Mass Center
- Every system of particles has a unique mass center (CM).
- CM's translational motion characterizes the system's overall motion.
- All the mass of a rigid body or particle system can be conceptually concentrated at its CM.
- CM location is closer to more massive particles.
- Applying a force to an extended body at its CM causes only translational motion.
- Applying a force at another point results in translation and rotation.
Center of Mass of a System of Discrete Particles
- A system of discrete particles has finite distances between its particles.
- For particles with masses m1, m2...mi...mn, velocities v1, v2...vi...vn, and positions r1, r2...ri...rn, the center of mass (r) is:
- r = (m1r1 + m2r2 + ...+ mnrn ) / (m1 + m2 + ...+ mn)
- r = Σmiri / Σmi
- r = Σmiri / M
- Where M = Σmi is the total mass of the system
- The Cartesian coordinates of the center of mass are:
- x = Σmixi / M
- y = Σmiyi / M
- z = Σmizi / M
Center of Mass of a System of Two Particles
- The CM of two particles lies on the line connecting them.
- It divides the distance between the particles inversely proportional to their masses.
- If masses are equal, the CM is at the midpoint.
Center of Mass of an Extended Body
- An extended body has a continuous mass distribution.
- CM of an extended body is determined by:
- r = (∫ r dm ) / ∫ dm
- r = (∫ r dm ) / M
- Where M is the total mass of the body.
- The CM of a symmetrical uniform body lies on its axis of symmetry.
Motion of the Center of Mass
- A system's motion is a combination of CM translation and the motion of particles relative to the CM.
- The system's total linear momentum equals the momentum due to CM translation.
- CM is useful for analyzing the system's overall translation.
Center of Mass Frame of Reference
- The centroidal frame is fixed to the CM of the system, moving with it.
- In this frame, CM's position, velocity, and acceleration are zero.
- The sum of mass moments vanishes in this frame.
Impulse-Momentum Principle
- Linear impulse (Imp) is the time integral of force:
- Imp = ∫ F dt
- For constant force: Imp = F Δt
- Linear momentum (p) is the product of mass and velocity:
- p = mv
-
Impulse-Momentum Principle: The momentum change of a body equals the impulse of the net force acting on it over a given time interval.
- Imp = Δp = p2 - p1
- Where p2 is momentum at time t2, and p1 is momentum at time t1.
- Conservation of Linear Momentum: If the net impulse of external forces on a system is zero over a time interval, the system's total linear momentum remains constant.
Impulsive Motion
- An impulsive force is a large force acting for a very short time.
- Impulsive motion describes a body's motion under the action of an impulsive force.
- External forces negligible compared to the impulsive force are considered non-impulsive.
- Non-impulsive forces are often disregarded when analyzing impulsive motion.
Motion of the Center of Mass in One Dimension
- Consider two bodies with different initial velocities.
- The CM velocity is the weighted average of their velocities: (m1v1 + m2v2) / (m1 + m2).
- The velocities of each body in the centroidal frame are their initial velocities in the inertial frame minus the CM velocity.
- Centroidal Frame: Frame of reference fixed to the CM.
Motion of the Center of Mass in Vector Form
- The velocity of the CM for particles with different velocities is calculated using the vector sum of masses and velocities.
- The velocity of each particle in the centroidal frame is found by subtracting the CM velocity from the particle's velocity in the inertial frame.
Conservation of Linear Momentum for a System of Particles
- If internal forces are zero, the net external force on the system equals its total mass times its acceleration.
- For a system of particles:
- Σ Fi + Σ fi = mi a
- Σ fi = 0, so Σ Fi = M a
- Where:
- Σ Fi is the sum of external forces on the particles.
- Σ fi is the sum of internal forces on the particles.
- M is the total mass of the system.
- The CM of the system accelerates as if it were a single particle with mass M subjected to the net external force.
Application of Newton's Laws of Motion and Momentum to a System of Particles
- Newton's third law (action-reaction) extends to systems of particles.
Simple Atwood Machine
- This consists of two masses connected by a string over a pulley.
- CM acceleration is determined by the conservation of linear momentum.
- Tension in the string is found using the force equation.
Systems of Particles with a Constant Force
- Consider two blocks connected by an unstretched spring, with a constant force applied to one block.
- CM acceleration is equal to the force divided by the system's total mass.
- Spring extension is found using the spring's equation of motion.
Work-Energy Methods for a System of Particles
- The work-energy theorem can be applied to individual particles or the whole system.
- Potential energy is useful for calculating work done by internal forces.
- If the total work by internal and external forces is zero, the system's mechanical energy is conserved.
Conservation of Mechanical Energy
- The system's total mechanical energy remains constant if the work done by internal forces (not described by a potential energy equation) is zero, and no external forces act.
- Total mechanical energy changes according to the work done by external forces.
Impulsive Forces
- Impulsive forces cause changes in a system's momentum.
- Conservation of momentum states that total momentum is constant when no external impulsive forces act.
- The impulse of external forces acting on a system over a time interval equals the system's momentum change.
Variable Mass Systems
- A variable mass system gains or loses mass.
- Thrust force: The force exerted by mass being added or ejected from a system. It equals the rate of mass change multiplied by the relative velocity of the mass.
-
Rocket Propulsion:
- The motion of a rocket is analyzed by considering the force due to exhaust gas ejection.
- Rocket velocity is determined using conservation of momentum.
- Final rocket velocity depends on initial velocity, fuel ejection velocity, and the rocket's mass ratio (final mass divided by initial mass).
Oblique Collisions
- Oblique collision: Collision where velocity vectors are not aligned with the impact line.
- Analyze by resolving velocities into components along the tangent (t) direction and normal (n) direction to the impact line.
- Conservation of momentum: Momentum component along the tangent is conserved.
- Coefficient of restitution: Determines the relative speed of the bodies after collision along the normal direction.
Perfectly Inelastic Collisions
- In a perfectly inelastic collision, the bodies stick together after colliding.
- Coefficient of restitution for perfectly inelastic collisions is zero.
Perfectly Elastic Collisions
- Perfectly elastic collision: Kinetic energy is conserved.
- The coefficient of restitution is 1.
Kinetic Energy in Perfectly Elastic Impact
- Kinetic energy of a particle system is the sum of individual particle kinetic energies.
- In a perfectly elastic collision, kinetic energy is conserved before and after impact.
Inelastic Collisions
- In an inelastic collision, some kinetic energy is lost.
- In a perfectly inelastic collision, kinetic energy is lost as heat and sound.
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Description
This quiz delves into the fundamental concepts of kinematics and kinetics, exploring the principles of motion without forces and the effects of forces on motion. You will learn about Newton's Laws, work-energy methods, and how internal and external forces interact in a system of particles.