small force for a long time. 12 **Conservation** **of** linear **momentum** The **Law** **of** **Conservation** **of** Linear **Momentum** states: The total linear **momentum** **of** an isolated system is constant. Isolated implies no external force: 13 **Conservation** **of** linear **momentum**, cont’d This **law** helps us deal with collisions. If the system’s **momentum** can not change, the **momentum** before the collision must equal that after the collision. 14 **Conservation** **of** linear **momentum**, cont’d We can write this as/

). The motorcycle (with rider) has a mass **of** 350 kg. Calculate and compare the **momentum** **of** the car and motorcycle. **Conservation** **of** **Momentum** The **law** **of** **conservation** **of** **momentum** states when a system **of** interacting objects is not influenced by outside forces (like friction), the total **momentum** **of** the system cannot change. If you throw a rock forward from a skateboard, you will move backward in response. **Conservation** **of** **Momentum** Collisions in One Dimension A collision occurs/

://en.wikipedia.org/wiki/Physical_conservation_la w http://en.wikipedia.org/wiki/Physical_conservation_la w **Conservation** **of** **Momentum** The total **momentum** **of** interacting objects cannot change unless an external force is acting on them Interacting objects exchange **momentum** through equal and opposite forces What is **Momentum** Really? **Momentum** is the thing that is **conserved** (does not change) because the **laws** **of** nature (physics) are the same here, there, and over there (space invariant/

+ m s )gd Ballistic Pendulum **Momentum** conservationenergy **conservation** mv ob = ( m b + m p )v bp ½ (m b + m p )v bp 2 = (m b + m s )g y Impulse Change in **Momentum** Stiff Spring Soft Spring Rubber Bumper Magnetic bumper Force as a function **of** time graph analysis Rocket Thrust Analysis Newtons Third **Law** Connection to Change in **momentum** F t t Impulse = J = F/

is another way to say **momentum** is **conserved**! **Law** **of** **conservation** **of** linear **momentum** 9-5 **Conservation** **of** Linear **Momentum** © 2014 John Wiley & Sons, Inc. All rights reserved. 9-5 **Conservation** **of** Linear **Momentum** Check components **of** net external force to determine if you should apply **conservation** **of** **momentum** © 2014 John Wiley & Sons, Inc. All rights reserved. 9-5 **Conservation** **of** Linear **Momentum** Internal forces can change momenta **of** parts **of** the system, but cannot change the linear **momentum** **of** the entire system/

2f In component form, the total momenta in each direction are independently **conserved** p ix = p fx p iy = p fy p iz = p fz **Conservation** **of** **momentum** can be applied to systems with any number **of** particles **Conservation** **of** **Momentum**, Archer Example The archer is standing on a frictionless surface (ice) Approaches: Newton’s Second **Law** – no, no information about F or a Energy approach – no, no information/

BE **CONSERVED** DURING ELASTIC COLLISIONS 47 MECHANICS **MOMENTUM** VIDEO 48 MECHANICS WESBURY COLLEGE **OF** SCIENCE LEARNERS MODULE 1 p45-46 p47 p48 AKT 6. VRAE 1-4 49 KNOWLEDLEDGE AREA MECHANICS NEWTON’S SECOND **LAW** **OF** MOTION NEWTON’S FIRST **LAW** **OF** MOTION GR 11 MECHANICS 50 VIDEO NEWTON’S THREE **LAWS** **OF** MOTION 51 NEWTON’S SECOND **LAW** **OF** MOTION MATHEMATICAL EXPRESSION **OF** NEWTON’S SECOND **LAW** **OF** MOTION 52 NEWTON’S SECOND **LAW** **OF**/

total **momentum** **of** an isolated system equals its initial **momentum** 8 **Conservation** **of** **Momentum**, 2 **Conservation** **of** **momentum** can be expressed mathematically in various ways In component form, the total momenta in each direction are independently **conserved** p ix = p fx p iy = p fy p iz = p fz **Conservation** **of** **momentum** can be applied to systems with any number **of** particles This **law** is the mathematical representation **of** the **momentum** version **of** the isolated system model 9 **Conservation** **of** **Momentum**/

KEY CONCEPT SUMMARY KEY CONCEPT SUMMARY Forces transfer **momentum**. collision **conservation** **of** **momentum** 11.4 CHAPTER RESOURCES CHAPTER RESOURCES VOCABULARY KEY CONCEPT CHAPTER HOME A **law** stating that the amount **of** **momentum** a system **of** objects has does not change as long as there are no outside forces acting on that system. **conservation** **of** **momentum** KEY CONCEPT SUMMARY KEY CONCEPT SUMMARY Forces transfer **momentum**. collision **conservation** **of** **momentum** **momentum** 11.4 CHAPTER RESOURCES CHAPTER RESOURCES VOCABULARY/

the system Key Concept: In a closed system, the loss **of** **momentum** **of** one object equals the gain in **momentum** **of** another object—**momentum** is **conserved**. Key Concept: In a closed system, the loss **of** **momentum** **of** one object equals the gain in **momentum** **of** another object—**momentum** is **conserved**. Section 12.3 Newton’s Third **Law** **of** Motion and **Momentum** **Law** **of** **Conservation** **of** **Momentum** **Law** **of** **Conservation** **of** **Momentum** The total **momentum** in a group **of** objects doesn’t change unless outside forces act on the objects/

a theory to ensure that **conservation** **of** energy holds. (e.g., Energy in EM field). Consistent with experimental facts! **Conservation** **Laws** & Symmetry Principles (not in text!) In all **of** physics (not just mechanics) it can be shown: –Each **Conservation** **Law** implies an underlying symmetry **of** the system. –Conversely, each system symmetry implies a **Conservation** **Law**: Can show: Translational Symmetry Linear **Momentum** **Conservation** Rotational Symmetry Angular **Momentum** **Conservation** Time Reversal Symmetry Energy/

-diagonal members that represent tangential stresses. THE **CONSERVATION** **LAWS** – **momentum** **conservation** In 2D problems stress tensor has the shape: Writing the Taylor series with the inclusion **of** only the first member one holds (for x-direction): THE **CONSERVATION** **LAWS** – **momentum** **conservation** Substitution in Newton second **law**-axiom (**conservation** **of** **momentum**) for x-direction holds : or in vector form for 3D problem: or by components : THE **CONSERVATION** **LAWS** – **momentum** **conservation** The main problem is that we have/

also tells us that the total **momentum** **of** an isolated system equals its initial **momentum** **Conservation** **of** **Momentum**, 2 **Conservation** **of** **momentum** can be expressed mathematically in various ways In component form, the total momenta in each direction are independently **conserved** **Conservation** **of** **momentum** can be applied to systems with any number **of** particles **Conservation** **of** **Momentum**, Archer Example The archer is standing on a frictionless surface (ice) Approaches: Newton’s Second **Law** – no, no information about F/

2 v 1 D.m 1 v 1 2 + m 2 v 2 2 = (m 1 + m 2 )v 3 2 SECTION 1 1.2 Section Check Reason: By the **law** **of** **conservation** **of** **momentum**, we know that the total **momentum** before a collision is equal to the total **momentum** after a collision. SECTION 1 1.2 Section Check Answer Reason: Before the collision, the car/

· m/s north **Momentum** **Conservation** **of** **Momentum** The **law** **of** **conservation** **of** **momentum**: –The total **momentum** **of** objects that collide is the same before and after the collision. **Conservation** **of** **Momentum** According to the **law** **of** **conservation** **of** **momentum**; if one ball swings in then how many balls will swing out? **Conservation** **of** **Momentum** When the cue ball hits another ball, the motion **of** both balls change. The cue ball slows down and may change direction, so its **momentum** increases. **Conservation** **of** **Momentum** It seems as/

m/s 4132.9736 Alternate Solution 5168 40 1500 p Shown are **momentum** vectors (in g m/s). The black vector is the total **momentum** before the collision. Because **of** **conservation** **of** **momentum**, it is also the total **momentum** after the collisions. We can use trig to find its magnitude and direction. **Law** **of** Sines : 40 sin 1500 **Law** **of** Cosines : p 2 = 5168 2 + 1500 2 - 2 5168 1500 cos/

acting on the particle. This is equivalent to Newton’s Second **Law**. This is identical in form to the **conservation** **of** energy equation. This is the most general statement **of** the principle **of** **conservation** **of** **momentum** and is called the **conservation** **of** **momentum** equation. This form applies to non-isolated systems. This is the mathematical statement **of** the non-isolated system (**momentum**) model. Section 9.3 More About Impulse Impulse is a vector/

acting on the particle. This is equivalent to Newton’s Second **Law**. This is identical in form to the **conservation** **of** energy equation. This is the most general statement **of** the principle **of** **conservation** **of** **momentum** and is called the **conservation** **of** **momentum** equation. This form applies to non-isolated systems. This is the mathematical statement **of** the non-isolated system (**momentum**) model. Section 9.3 More About Impulse Impulse is a vector/

on the system **Law** **of** **conservation** **of** **momentum**-**law** stating that the total **momentum** **of** a system does not change if no net force acts on the system Key Concept: In a closed system, the loss **of** **momentum** **of** one object equals the gain in **momentum** **of** another object—**momentum** is **conserved**. Key Concept: In a closed system, the loss **of** **momentum** **of** one object equals the gain in **momentum** **of** another object—**momentum** is **conserved**. **Conservation** **of** **Momentum** Figure 17A and 17B **Conservation** **of** **Momentum** Figure 17C

Systems The previous part **of** Lesson 2 focused on the **Law** **of** **Conservation** **of** **Momentum**. It was stated that...previous part **of** Lesson 2 For a collision occurring between object 1 and object 2 in an isolated system, the total **momentum** **of** the two objects before the collision is equal to the total **momentum** **of** the two objects after the collision. That is, the **momentum** lost by object 1 is equal to/

the total **momentum** **of** an isolated system equals its initial **momentum** **Conservation** **of** **Momentum**, 2 **Conservation** **of** **momentum** can be expressed mathematically in various ways In component form, the total momenta in each direction are independently **conserved** p ix = p fx p iy = p fy p iz = p fz **Conservation** **of** **momentum** can be applied to systems with any number **of** particles This **law** is the mathematical representation **of** the **momentum** version **of** the isolated system model **Conservation** **of** **Momentum**, Archer/

example, if N z (e) = 0, L z is **conserved**. Linear & Angular **Momentum** **Conservation** **Laws**: –**Conservation** **of** Linear **Momentum** holds if internal forces obey the “Weak” **Law** **of** Action and Reaction: Only Newton’s 3 rd **Law** F ji = - F ij is required to hold! –**Conservation** **of** Angular **Momentum** holds if internal forces obey the “Strong” **Law** **of** Action and Reaction: Newton’s 3 rd **Law** F ji = - F ij holds, PLUS the forces must be/

that the total **momentum** **of** an isolated system equals its initial **momentum** 9 **Conservation** **of** **Momentum**, 2 **Conservation** **of** **momentum** can be expressed mathematically in various ways In component form, the total momenta in each direction are independently **conserved** **Conservation** **of** **momentum** can be applied to systems with any number **of** particles 10 11 12 13 **Conservation** **of** **Momentum**, Archer Example The archer is standing on a frictionless surface (ice) Approaches: Newton’s Second **Law** – no, no information/

Which is Rate **of** change **of** **momentum**. This is a more general statement **of** Newton’s second **law**. **Conservation** **of** **momentum** Principle **of** **conservation** **of** **momentum**: When two or more bodies interact, then the total **momentum** is **conserved** if no external forces act on the bodies. This can be derived from Newton’s third **law** (action/ reaction exerted on different bodies) It is a very important principle! Example – two balls colliding By **law** **of** **conservation** **of** **momentum**, **momentum** before collision is/

, and the “system” would have to include the table as well. The Cart and The Brick - Part BThe Cart and The Brick - Part B Proof **of** **Conservation** **of** **Momentum** The proof is based on Newton’s 3 rd **Law**. Whenever two objects collide (or exert forces on each other from a distance), the forces involved are an action-reaction pair, equal in strength, opposite in/

and columns the original matrices have (called the "size", "order" or "dimension"), and specifying how the entries **of** the matrices generate the new matrix.binary operationmatricesNumbersrealcomplex numbersmultipliedelementary arithmetic"size", "order" or "dimension" **Momentum** **conservation** One **of** the most powerful **laws** in physics is the **law** **of** **momentum** **conservation**.**momentum** Collision A collision is an isolated event in which two or more moving bodies (colliding bodies) exert forces on each other/

ix = p fx p iy = p fy p iz = p fz **Conservation** **of** **momentum** can be applied to systems with any number **of** particles 6 **Law** **of** **Conservation** **of** Linear **Momentum**, 2 Whenever two or more particles in an isolated system interact, the total **momentum** **of** the system remains constant The **momentum** **of** the system is **conserved**, not necessarily the **momentum** **of** an individual particle because other particles in the system may be interacting with it/

when no net external forces are applied to a system **of** two bodies, the total **momentum** **of** the system before is the same as the total **momentum** **of** the system after. 9 **Momentum** in 2-Dimensions The **Law** **of** **Conservation** **of** **Momentum** **Law** **of** **conservation** **of** linear **momentum** If there are no net external forces acting on a system **of** bodies then the total linear **momentum** **of** the system is **conserved**. NB.(1) No net external forces means an isolated/

: collisions, mass, initial velocities Results Next Slide Diagram Calculation Photo **Momentum** **Momentum** 1 **Conservation** **of** **momentum** Definition **of** **momentum** : Mass velocity (mv) (vector! Why?) **Conservation** **of** total **momentum** in the experiment Unit : Next Slide **Momentum** **Momentum** 2 **Conservation** **of** **momentum** **Law** **of** **conservation** **of** **momentum** In a collision the total **momentum** **of** the objects before the collision is equal to the total **momentum** after the collision, provided that there is no external force acting on/

to explain many safety technologies ex. road barriers 3.9 **Conservation** **of** **Momentum** **Law** **of** **Conservation** **of** **Momentum** if the net force acting on a system is zero, the sum **of** the **momentum** before an interaction equals the **momentum** after the interaction Summomentum Since p = mv, mv can be substituted for p Action Reaction How does Newton’s Third **Law** apply to **conservation** **of** **momentum**? each action has an opposite and equal reaction Forces act/

a crystal – space is no longer uniform but has a new symmetry – its periodic, so the **law** **of** **conservation** **of** **momentum** is replaced by a new **law** – the **conservation** **of** ‘crystal **momentum**’ in which **momentum** is **conserved** to within a factor **of** ħG. E.g. Diffraction: wavevector allowed to change by factors **of** G Phonon creation: Adding a G vector to a phonon’s wavevector does not change its properties, but its crystal/

**of** **Momentum** 12.2 Newton’s First and Second **Laws** **of** Motion In each collision, the total **momentum** **of** the train cars does not change—**momentum** is **conserved**. **Conservation** **of** **Momentum** 12.2 Newton’s First and Second **Laws** **of** Motion In each collision, the total **momentum** **of** the train cars does not change—**momentum** is **conserved**. **Conservation** **of** **Momentum** 12.2 Newton’s First and Second **Laws** **of** Motion In each collision, the total **momentum** **of** the train cars does not change—**momentum** is **conserved**. **Conservation** **of** **Momentum**/

the **law** **of** **conservation** **of** **momentum**, if no net force acts on a system, then the total **momentum** **of** the system does not change. **Conservation** **of** **Momentum** 12.3 Newton’s Third **Law** **of** Motion and **Momentum** In each collision, the total **momentum** **of** the train cars does not change—**momentum** is **conserved**. **Conservation** **of** **Momentum** 12.3 Newton’s Third **Law** **of** Motion and **Momentum** In each collision, the total **momentum** **of** the train cars does not change—**momentum** is **conserved**. **Conservation** **of** **Momentum** 12.3 Newton’s Third **Law** **of**/

usually starts out slowly and then gets faster and faster. This may appear to be a violation **of** the **law** **of** **conservation** **of** angular **momentum** but is in fact a beautiful example **of** its validity. **Conservation** **of** Angular **Momentum** Angular **momentum** is the product **of** the rotational inertia and the rotational speed and, in the absence **of** a net torque, remains constant. Therefore, if the rotational inertia decreases, the rotational speed must increase/

acting on the particle. This is equivalent to Newton’s Second **Law**. This is identical in form to the **conservation** **of** energy equation. This is the most general statement **of** the principle **of** **conservation** **of** **momentum** and is called the **conservation** **of** **momentum** equation. This form applies to non-isolated systems. This is the mathematical statement **of** the non-isolated system (**momentum**) model. Section 9.3 More About Impulse Impulse is a vector/

new force acting on the particle – –This is equivalent to Newton’s Second **Law** –This is identical in form to the **conservation** **of** energy equation –This is the most general statement **of** the principle **of** **conservation** **of** **momentum** and is called the **conservation** **of** **momentum** equation This form applies to non-isolated systems –This is the mathematical statement **of** the non- isolated system More About Impulse Impulse is a vector quantity The/

**momentum** Section 2 **Conservation** **of** **Momentum** Chapter 6 **Momentum** is **Conserved** The **Law** **of** **Conservation** **of** **Momentum**: The total **momentum** **of** all objects interacting with one another remains constant regardless **of** the nature **of** the forces between the objects. m1v1,i + m2v2,i = m1v1,f + m2v2,f total initial **momentum** = total final **momentum** **Conservation** **of** **Momentum** Section 2 **Conservation** **of** **Momentum** Chapter 6 **Conservation** **of** **Momentum** Chapter 6 Sample Problem **Conservation** **of** **Momentum** Section 2 **Conservation** **of** **Momentum**/

m2 u2 m1 v1 m2 v2 Principle **of** **conservation** **of** **momentum** The **momentum** **of** m1 time **of** action m1.u1 m1v1 time m1 u1 m2 u2 m1 v1 m2 v2 Principle **of** **conservation** **of** **momentum** The **momentum** **of** m2 time **of** action m2.v2 m2.u2 time m1 u1 m2 u2 m1 v1 m2 v2 Principle **of** **conservation** **of** **momentum** For N bodies in collision. Without external force, sum **of** momenta before = sum **of** momenta after Collisions in 2-dimension/

energy from the agent doing the work Lecture 9 Spring 2008 4/23/2008 Energy is **Conserved**! The total energy (in all forms) in a “closed” system remains constant This is one **of** nature’s “**conservation** **laws**” **Conservation** applies to: Energy (includes mass via E = mc2) **Momentum** Angular **Momentum** Electric Charge **Conservation** **laws** are fundamental in physics, and stem from symmetries in our space and time Emmy Noether formulated/

with an initial velocity v1. The mass collides with a 5-kg mass m2, which is initially at rest. Find the final velocity **of** the masses after the collision if it is perfectly inelastic. According to the **Law** **of** **Conservation** **of** **Momentum** example An archer stands at rest on frictionless ice and fires a 0.5-kg arrow horizontally at 50.0 m/s. The combined/

ALSO equal zero. If delta p equals zero for a system, then **momentum** is **CONSERVED**. You must be able to describe the RECOIL EFFECT in BOTH **of** the following ways: By Newton’s third **law**, if spacecraft pushes gasses out backwards, gasses push spacecraft forwards. (3 points) By **conservation** **of** **momentum**, say… Total initial **momentum** equals zero; **momentum** **of** gasses one way + spacecraft other way (which are equal and opposite/

root, and we get mv = constant. The speed changed to relative quantity, which has now a direction. Thus the **conservation** **law** **of** **momentum** in three-dimensional space can be united to the **conservation** **law** **of** absolute **momentum** at low relative speeds. For the moving bodies there exists only one single **law** **of** **conservation**. Lets write m² c² = constant for a body, which moves at absolute speed c in relation to rest/

same. The equation you see here states this fact mathematically. The vector sum **of** the momenta **of** all the objects at an initial time equals the total **momentum** at any later (final) time. **Conservation** **of** **momentum** The **law** **of** **conservation** **of** **momentum** can be derived from Newton’s 2nd and 3rd **laws** Newton’s 2nd **law** F = ma Newton’s 3rd **law** Forces are equal but opposite * Refer to Kinetic Books- 8.7 For/

y directions After Before **Conservation** **of** **Momentum** External force Internal forces Isolated system: External force = 0 Newton’s 2nd **law** Newton’s 3rd **law** You can add these forces to obtain the net (internal) force for the system The sum **of** these forces cannot be applied to a single object. **Conservation** **of** **Momentum** Total **momentum** **of** an isolated system is **conserved**. Total **momentum** **Momentum** **conservation** gives a vector equation. Note that **momentum** **conservation** is valid for an/

1 + m 2 Δv 2 = 0 **Conservation** **of** Linear **Momentum**.**Conservation** **of** Linear **Momentum**. UCSD Physics 10 Winter 20066 Linear **Momentum** Often misused word, though most have the right ideaOften misused word, though most have the right idea **Momentum**, denoted p, is mass times velocityMomentum, denoted /!!! –Newton’s 3 rd **Law** **of** Motion! Which Canoe experience the greater acceleration?Which Canoe experience the greater acceleration? How do the momenta **of** each canoe compare?How do the momenta **of** each canoe compare? –Be /

this experiment. **Conservation** **of** **Momentum** - - Newton’s Third **Law** **Conservation** **of** **Momentum** **Law** **of** the **Conservation** **of** **Momentum** (mass in motion): The total **momentum** is **conserved** (or does not change) for any group **of** objects, unless outside forces (such as friction) act on the objects. Relationship between **Momentum**, Mass, & Velocity What causes an object’s **momentum** to increase? How does velocity affect **momentum**? How does mass affect **momentum**? **Momentum** = m x v Calculating **Momentum** - Newton’s Third **Law** Which has/

is zero, neither its velocity nor its **momentum** change. Because **momentum** is the product **of** mass and velocity, the force on an object equals its change in **momentum**. Lesson 4-4 **Conservation** **of** **Momentum** According to the **law** **of** **conservation** **of** **momentum**, the total **momentum** **of** a group **of** objects stays the same unless outside forces such as friction act on the objects. What is the **law** **of** **conservation** **of** **momentum**? Lesson 4-5 **Conservation** **of** **Momentum** (cont.) When colliding objects bounce off/

is zero, neither its velocity nor its **momentum** change. Because **momentum** is the product **of** mass and velocity, the force on an object equals its change in **momentum**. Lesson 4-4 **Conservation** **of** **Momentum** According to the **law** **of** **conservation** **of** **momentum**, the total **momentum** **of** a group **of** objects stays the same unless outside forces such as friction act on the objects. What is the **law** **of** **conservation** **of** **momentum**? Lesson 4-5 **Conservation** **of** **Momentum** (cont.) When colliding objects bounce off/

–3. For every force there is an equal and opposite reaction force **Conservation** **Laws** in Astronomy Our goals for learning: Why do objects move at constant velocity if no force acts on them? What keeps a planet rotating and orbiting the Sun? Where do objects get their energy? **Conservation** **of** **Momentum** The total **momentum** **of** interacting objects cannot change unless an external force is acting on them/

-production must not violate some very fundamental **laws** in physics: Charge **conservation**, total linear **momentum**, total relativistic energy are to be obeyed in the process Due to kinematical consideration (energy and linear **momentum** **conservations**) pair production cannot occur in empty space Must occur in the proximity **of** a nucleus Will see this in an example Energy threshold Due to **conservation** **of** relativistic energy, pair production can only occur/

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