Kinematics and Kinetics Overview

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Questions and Answers

What determines the final velocity of a rocket?

Rocket's initial mass, fuel type, and exit pressure

Fuel ejection velocity, rocket's cross-sectional area, and thrust

Initial velocity, fuel ejection velocity, and mass ratio (correct)

Initial velocity, fuel ejection speed, and rocket's fuel type

In an oblique collision, which component of momentum is conserved?

Only the tangent component (correct)

Neither the tangent nor the normal component

Only the normal component

Both tangent and normal components

What characterizes a perfectly inelastic collision?

There is no momentum change

Kinetic energy is conserved

The two bodies bounce off each other

The two bodies stick together after colliding (correct)

What is the coefficient of restitution for perfectly elastic collisions?

1 Signup and view all the answers

What happens to kinetic energy in a perfectly inelastic collision?

Some of it is lost, typically as heat and sound Signup and view all the answers

What is the primary focus of kinematics?

Study of motion without considering forces Signup and view all the answers

Which principle is the impulse and momentum method developed from?

Newton's second law Signup and view all the answers

What defines an internal force within a system of particles?

A force that acts between particles within the system Signup and view all the answers

How does the center of mass behave when a force is applied at it?

Only translational motion occurs Signup and view all the answers

What is an example of an extended body?

A system with an infinite number of densely packed particles Signup and view all the answers

Which of the following accurately describes a rigid body?

Particle separations remain unchanged Signup and view all the answers

What characterizes the center of mass of a system of discrete particles?

It represents the average location of all particles' mass Signup and view all the answers

Which statement is true about external forces acting on a system of particles?

They can change the motion of the mass center Signup and view all the answers

What is linear impulse defined as?

The integral of force over time. Signup and view all the answers

How is linear momentum calculated?

By multiplying mass and velocity. Signup and view all the answers

What does the Impulse-Momentum Principle state?

The change in momentum equals the net impulse over a time interval. Signup and view all the answers

What characterizes impulsive motion?

It occurs under the influence of large forces for short durations. Signup and view all the answers

What is the role of non-impulsive forces during impulsive motion?

They are usually neglected. Signup and view all the answers

How is the velocity of the center of mass calculated for a system of two bodies?

By dividing their total momentum by the sum of their masses. Signup and view all the answers

In vector form, how is the velocity of the center of mass determined for a system of particles?

Through the vector sum of masses and velocities. Signup and view all the answers

What is the condition for the conservation of linear momentum in a system of particles?

The net external force acting on the system must be zero. Signup and view all the answers

What is the relationship between the center of mass (CM) of a system and the net external force acting on it?

The CM accelerates as if it were a single particle with mass equal to the total mass of the system. Signup and view all the answers

Which statement about Newton's third law in the context of systems of particles is correct?

It states that every action has an equal and opposite reaction. Signup and view all the answers

How can the acceleration of the center of mass in a system of two blocks connected by an unstretched spring be determined?

By using the formula acceleration = force / total mass. Signup and view all the answers

What happens to the total mechanical energy of a system when no external forces act and the internal forces do no work?

It remains constant. Signup and view all the answers

Which of the following correctly defines impulsive forces?

Forces that lead to a change in the momentum of a system. Signup and view all the answers

What defines a variable mass system?

A system where mass is ejected or added. Signup and view all the answers

To calculate the work done by internal forces in a system, which concept is crucial?

Potential energy consideration. Signup and view all the answers

What does the principle of conservation of momentum imply for a system with no external impulsive forces?

The total momentum remains constant. Signup and view all the answers

How is the center of mass calculated for a system of discrete particles?

r = Σmiri / Σmi Signup and view all the answers

What does the center of mass of a two-particle system depend on?

The inverse ratio of their masses Signup and view all the answers

In the Cartesian coordinates, how is the x-coordinate of the center of mass expressed?

x = Σmixi / M Signup and view all the answers

What is the equation for the center of mass (CM) of an extended body?

r = (∫ r dm) / ∫ dm Signup and view all the answers

How does the motion of a system of particles relate to the center of mass?

The motion can be viewed as a superposition of CM translation and particle motion Signup and view all the answers

What is true about the centroidal frame of reference?

The sum of the mass moments in this frame vanishes Signup and view all the answers

Where does the center of mass of a uniform symmetrical body lie?

On the axis of symmetry of the body Signup and view all the answers

What happens to the center of mass when two equal masses are separated by a distance?

It is positioned at the midpoint between the two masses Signup and view all the answers

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...m_i...m_n, velocities v₁, v₂...v_i...v_n, and positions r₁, r₂...r_i...r_n, the center of mass (r) is:
- r = (m₁r₁ + m₂r₂ + ...+ m_nr_n ) / (m₁ + m₂ + ...+ m_n)
- r = Σm_ir_i / Σm_i
- r = Σm_ir_i / M
- Where M = Σm_i is the total mass of the system
The Cartesian coordinates of the center of mass are:
- x = Σm_ix_i / M
- y = Σm_iy_i / M
- z = Σm_iz_i / 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 F_i + Σ f_i = m_i a
- Σ f_i = 0, so Σ F_i = M a
- Where:
  - Σ F_i is the sum of external forces on the particles.
  - Σ f_i 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.