Table of Contents
Module 8. Work and Power .................................................................................. 2
Summary: ............................................................................................................................. 2
Formulas: ............................................................................................................................. 3
8.1 Work ............................................................................................................................... 5
8.2 Kinetic Energy and the Work–Energy Theorem ....................................................... 5
8.3 Work and Energy with Varying Forces ...................................................................... 6
8.4 Power ............................................................................................................................. 7
8.5 Gravitational Potential Energy .................................................................................... 7
8.6
Elastic Potential Energy ........................................................................................... 8
8.7 Conservative and Nonconservative Forces .............................................................. 9
8.8 Force and Potential Energy ....................................................................................... 10
Module 9. Momentum ........................................................................................ 11
Summary: ........................................................................................................................... 11
Formula: ............................................................................................................................. 12
9.1 Momentum and Impulse ........................................................................................... 13
9.2 Conservation of Linear Momentum ......................................................................... 14
9.3 Momentum Conservation and Collisions ................................................................ 14
9.4 Center of Mass ............................................................................................................ 15
References........................................................................................................... 15
Module 8. Work and Power
Summary:
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Work (W)- A scalar quantity with the SI unit Joule (J). 1 Joule is 1 Newton meter
(N ⋅ m).
Work can be positive, negative or zero.
When a particle undergoes a displacement, it speeds up if 𝑊𝑡𝑜𝑡 > 0, slows
down if 𝑊𝑡𝑜𝑡 < 0, and maintains the same speed if 𝑊𝑡𝑜𝑡 = 0
Kinetic Energy (K)- a scalar quantity that depends on the particle’s mass and
speed.
Work-Energy Theorem- work done by the net force on a particle equals the
change in the particle’s kinetic energy.
The kinetic energy of a particle is equal to the total work that was done to
accelerate it from rest to its present speed.
The kinetic energy of a particle is equal to the total work that particle can do in
the process of being brought to rest.
The total kinetic energy of a composite system can change, even though no
work is done by forces applied by objects that are outside the system.
Hooke’s Law- force is directly proportional to elongation for elongations that
are not too great.
Power- a scalar quantity, that describes the time rate at which work is done. SI
unit is watt (W). 1 W = 1 Joule/s
1 horsepower=746 W=0.746 kW
Potential Energy- a measure of the potential or possibility for work to be done,
energy that is associated with position.
Gravitational Potential Energy (𝑈𝑔𝑟𝑎𝑣 )- potential energy associated with an
object’s weight and its height above the ground.
The expression for gravitational potential energy is the same whether the
object’s path is curved or straight.
Elastic Potential Energy- energy stored in a deformable object such as a spring
or rubber band.
An object is called elastic if it returns to its original shape and size after being
deformed.
Conservative Forces- forces whose works are reversible
Nonconservative Forces- forces whose works are not reversible.
Internal Energy- The energy associated with the change in the state of the
materials.
Law of Conservation of Energy- energy is never created or destroyed; it only
changes form.
Formulas:
8.1 Work
•
𝑊 = 𝐹𝑠 cos 𝜙
• 𝑊 = 𝑚𝑔ℎ
• 𝑊𝑡𝑜𝑡 = ∑ 𝑊 = 𝐹𝑛𝑒𝑡 𝑠 cos 𝜙
8.2 Kinetic Energy and Work Energy Theorem
1
• 𝐾 = 𝑚𝑣 2
2
• 𝑊𝑡𝑜𝑡 = 𝐾2 − 𝐾1 = ∆𝐾
8.3 Work and Energy with Varying Forces
𝑥
• 𝑊 = ∫𝑥 2 𝐹𝑥 𝑑𝑥
1
• 𝐹𝑥 = 𝑘𝑥
1
• 𝑊 = 𝑘𝑋 2
2
1
• 𝑊 = 𝑘(𝑥22 − 𝑥12 )
2
1
• 𝑊𝑡𝑜𝑡 = 𝑚(𝑣22 − 𝑣12 )
• 𝑊=
2
𝑃2
∫𝑃 𝐹 cos 𝜙 𝑑𝑙
1
8.4 Power
• 𝑃𝑎𝑣 = 𝐹|| 𝑣𝑎𝑣 =
• 𝑃 = 𝐹|| 𝑣 =
∆𝑊
∆𝑡
𝑑𝑊
𝑑𝑡
8.5 Gravitational Potential Energy
•
•
𝑈𝑔𝑟𝑎𝑣 = 𝑚𝑔𝑦
𝑊𝑔𝑟𝑎𝑣 = 𝑚𝑔𝑦1 − 𝑚𝑔𝑦2 = −∆𝑈𝑔𝑟𝑎𝑣
•
1
2
1
𝑚𝑣12 + 𝑚𝑔𝑦1 + 𝑊𝑜𝑡ℎ𝑒𝑟 = 𝑚𝑣22 + 𝑚𝑔𝑦2
• 𝐸 = 𝐾 + 𝑈𝑔𝑟𝑎𝑣 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
2
8.6 Elastic Potential Energy
1
• 𝑈𝑒𝑙 = 𝑘𝑥 2
2
1
1
• 𝑊𝑡𝑜𝑡 = 𝑊𝑒𝑙 = 𝑘𝑥12 − 𝑘𝑥22
•
1
𝑚𝑣12
2
+
2
2
1
2
𝑘𝑥1 = 𝑚𝑣22
2
2
1
1
+ 𝑘𝑥22
2
• 𝐸 = 𝐾 + 𝑈𝑒𝑙
General Work Energy Theorem
• 𝐾1 + 𝑈𝑔𝑟𝑎𝑣,1 + 𝑈𝑒𝑙,1 + 𝑊𝑜𝑡ℎ𝑒𝑟 = 𝐾2 + 𝑈𝑔𝑟𝑎𝑣,2 + 𝑈𝑒𝑙,2
• 𝐸 =𝐾+𝑈
1
• 𝑈 = 𝑈𝑔𝑟𝑎𝑣 + 𝑈𝑒𝑙 = 𝑚𝑔𝑦 + 𝑘𝑥 2
2
8.7 Conservative and Nonconservative Forces
• ∆𝑈𝑖𝑛𝑡 = −𝑊𝑜𝑡ℎ𝑒𝑟
• 𝐾1 + 𝑈1 − ∆𝑈𝑖𝑛𝑡 = 𝐾2 + 𝑈2
8.8 Force and Potential Energy
• 𝐹𝑥 (𝑥) = −
• 𝐹𝑥 = −
• 𝐹𝑦 = −
• 𝐹𝑧 = −
𝑑𝑈(𝑥)
𝑑𝑥
𝜕𝑈
𝜕𝑥
𝜕𝑈
𝜕𝑦
𝜕𝑈
𝜕𝑧
𝜕𝑈
• ⃗𝑭 = − (
𝜕𝑥
𝑖̂ +
𝜕𝑈
𝜕𝑦
𝑗̂ +
𝜕𝑈
𝜕𝑧
⃗𝑈
𝑘̂) = −𝛁
Note: partial derivatives are derivatives where all other variables aside from the
variable taken with respect to is considered a constant.
Legend:
𝑊-work
𝜙- angle
𝑊𝑡𝑜𝑡 - total work
𝐾- kinetic energy
𝑘- spring constant
𝑃𝑎𝑣 - average power
𝑃- instantaneous power
𝐹|| - force that acts on the particle
𝑈𝑔𝑟𝑎𝑣 = gravitational potential energy
𝑊𝑔𝑟𝑎𝑣 - work by gravity
𝑊𝑜𝑡ℎ𝑒𝑟 - work by other
𝐸- total mechanical energy
𝑈𝑒𝑙 - elastic potential energy
∆𝑈𝑖𝑛𝑡 - change in internal energy
8.1 Work
Work (W)- A scalar quantity with the SI unit Joule (J). 1 Joule is 1 Newton meter (N ⋅ m)
The work done by a constant force is given by the product of the magnitude of the
force and the magnitude of the displacement (s).
𝑊 = 𝐹𝑠
To lift an object against gravity for a certain height (h), the work is equal to
𝑊 = 𝑚𝑔ℎ
If the force is at an angle with the displacement, the work is given by:
𝑊 = 𝐹𝑠 cos 𝜙
Work can be positive, negative or zero.
The total work (𝑊𝑡𝑜𝑡 ) done an object can be the sum of the works done by the
individual forces or by using the net force on the object as the force in the given
formula
𝑊𝑡𝑜𝑡 = ∑ 𝑊 = 𝐹𝑛𝑒𝑡 𝑠 cos 𝜙
8.2 Kinetic Energy and the Work–Energy Theorem
When a particle undergoes a displacement, it speeds up if 𝑊𝑡𝑜𝑡 > 0, slows down if
𝑊𝑡𝑜𝑡 < 0, and maintains the same speed if 𝑊𝑡𝑜𝑡 = 0
Kinetic Energy (K)- a scalar quantity that depends on the particle’s mass and speed.
1
𝐾 = 𝑚𝑣 2
2
Work-Energy Theorem- work done by the net force on a particle equals the change in
the particle’s kinetic energy.
𝑊𝑡𝑜𝑡 = 𝐾2 − 𝐾1 = ∆𝐾
The kinetic energy of a particle is equal to the total work that was done to accelerate
it from rest to its present speed.
The kinetic energy of a particle is equal to the total work that particle can do in the
process of being brought to rest.
The total kinetic energy of a composite system can change, even though no work is
done by forces applied by objects that are outside the system.
𝑥2
𝑊 = ∫ 𝐹𝑥 𝑑𝑥
𝑥1
8.3 Work and Energy with Varying Forces
Work done by a varying force in a straight-line motion is given by:
The force needed to keep a string stretched is given by:
𝐹𝑥 = 𝑘𝑥
Where k is the force constant/spring constant of the spring.
Hooke’s Law- force is directly proportional to elongation for elongations that are not
too great.
The work needed to elongate the spring is given by (assuming spring starts
unstretched):
1
𝑊 = 𝑘𝑋 2
2
If the string is already stretched:
𝑥2
1
1
1
𝑊 = ∫ 𝐹𝑥 𝑑𝑥 = 𝑘𝑥22 − 𝑘𝑥12 = 𝑘(𝑥22 − 𝑥12 )
2
2
2
𝑥1
Work-Energy Theorem for Straight Line Motion (Varying Forces)
1
1
1
𝑊𝑡𝑜𝑡 = 𝑚𝑣22 − 𝑚𝑣12 = 𝑚(𝑣22 − 𝑣12 )
2
2
2
Work–Energy Theorem for Motion Along a Curve
The work done by a force on a particle as it moves from 𝑃1 to 𝑃2 is
𝑃2
𝑊 = ∫ 𝐹 cos 𝜙 𝑑𝑙
𝑃1
8.4 Power
Power- a scalar quantity, that describes the time rate at which work is done. SI unit is
watt (W). 1 W = 1 Joule/s
1 ℎ𝑜𝑟𝑠𝑒𝑝𝑜𝑤𝑒𝑟 = 746 𝑊 = 0.746 𝑘𝑊
•
Average Power (𝑃𝑎𝑣 )
𝑃𝑎𝑣 = 𝐹|| 𝑣𝑎𝑣 =
•
∆𝑊
∆𝑡
Instantaneous Power (𝑃)
𝑃 = 𝐹|| 𝑣 =
𝑑𝑊
𝑑𝑡
8.5 Gravitational Potential Energy
Potential Energy- a measure of the potential or possibility for work to be done,
energy that is associated with position.
Gravitational Potential Energy (𝑈𝑔𝑟𝑎𝑣 )- potential energy associated with an object’s
weight and its height above the ground.
𝑈𝑔𝑟𝑎𝑣 = 𝑚𝑔𝑦
The work done by gravity is
𝑊𝑔𝑟𝑎𝑣 = 𝑚𝑔𝑦1 − 𝑚𝑔𝑦2 = −∆𝑈𝑔𝑟𝑎𝑣
Conversation of Total Mechanical Energy (Gravitational Forces only)
1
1
𝑚𝑣12 + 𝑚𝑔𝑦1 = 𝑚𝑣22 + 𝑚𝑔𝑦2
2
2
Total Mechanical Energy of the System (E) [if only gravity does work]
𝐸 = 𝐾 + 𝑈𝑔𝑟𝑎𝑣 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
If other forces other than gravity do work, then:
1
1
𝑚𝑣12 + 𝑚𝑔𝑦1 + 𝑊𝑜𝑡ℎ𝑒𝑟 = 𝑚𝑣22 + 𝑚𝑔𝑦2
2
2
Note: E decreases if 𝑊𝑜𝑡ℎ𝑒𝑟 is negative and increases if 𝑊𝑜𝑡ℎ𝑒𝑟 is positive.
The expression for gravitational potential energy is the same whether the object’s
path is curved or straight. [−𝑚𝑔∆𝑦 = −𝑚𝑔(𝑦2 − 𝑦1 ) = 𝑚𝑔𝑦1 − 𝑚𝑔𝑦2 ]
8.6Elastic Potential Energy
Elastic Potential Energy- energy stored in a deformable object such as a spring or
rubber band. An object is called elastic if it returns to its original shape and size after
being deformed.
1
𝑈𝑒𝑙 = 𝑘𝑥 2
2
For cases where only elastic potential energy does work
•
Total Work
1
1
𝑊𝑡𝑜𝑡 = 𝑊𝑒𝑙 = 𝑘𝑥12 − 𝑘𝑥22
2
2
•
Conservation of Total Mechanical Energy (Elastic Forces only)
1
1
1
1
𝑚𝑣12 + 𝑘𝑥12 = 𝑚𝑣22 + 𝑘𝑥22
2
2
2
2
•
Total Mechanical Energy of the system
𝐸 = 𝐾 + 𝑈𝑒𝑙
For cases where gravitational and elastic potential energy is doing work:
•
Work-Energy Theorem
𝐾1 + 𝑈𝑔𝑟𝑎𝑣,1 + 𝑈𝑒𝑙,1 + 𝑊𝑜𝑡ℎ𝑒𝑟 = 𝐾2 + 𝑈𝑔𝑟𝑎𝑣,2 + 𝑈𝑒𝑙,2
•
Total Mechanical Energy of the System
𝐸 =𝐾+𝑈
•
Where the potential energy (U) is the sum of gravitational potential energy and
elastic potential energy.
1
𝑈 = 𝑈𝑔𝑟𝑎𝑣 + 𝑈𝑒𝑙 = 𝑚𝑔𝑦 + 𝑘𝑥 2
2
8.7 Conservative and Nonconservative Forces
Conservative Forces
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•
•
•
Forces whose works are reversible
Examples:
o Gravitational Force
o Spring Force
Work done by a conservative force has the following properties:
o It can be expressed as the difference between the initial and final values
of a potential-energy function.
o It is reversible.
o It is independent of the path of the object and depends on only the
starting and ending points.
o When the starting and ending points are the same, the total work is
zero.
When the only forces that do work are conservative forces, the total
mechanical energy (E = K + U) is constant and conserved.
Nonconservative Forces
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•
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•
•
•
•
Forces whose works are not reversible.
The work done by a nonconservative force cannot be represented by a
potential-energy function.
Lost kinetic energy can’t be recovered.
Total mechanical energy is not conserved.
Dissipative Force- nonconservative force that causes mechanical energy to be
lost or dissipated.
There are also nonconservative forces that increase mechanical energy.
Example:
o Friction
o Fluid Resistance
Internal Energy
•
•
•
The energy associated with the change in the state of the materials.
Higher temperature means higher internal energy, and lower temperature
means lower internal energy.
Increase in the internal energy is exactly equal to the absolute value of the
work done by friction.
∆𝑈𝑖𝑛𝑡 = −𝑊𝑜𝑡ℎ𝑒𝑟
Note: ∆𝑈𝑖𝑛𝑡 is the change in internal energy
Law of Conservation of Energy (Energy is never created or destroyed; it only changes
form)
𝐾1 + 𝑈1 − ∆𝑈𝑖𝑛𝑡 = 𝐾2 + 𝑈2
8.8 Force and Potential Energy
Force can be expressed in potential energy:
𝐹𝑥 (𝑥) = −
𝑑𝑈(𝑥)
𝑑𝑥
Force and Potential Energy in Three Dimensions- the components of the force are
given by partial derivatives:
𝐹𝑥 = −
𝜕𝑈
𝜕𝑥
𝐹𝑦 = −
𝜕𝑈
𝜕𝑦
𝐹𝑧 = −
𝜕𝑈
𝜕𝑧
Note: partial derivatives assume that variables other than the variable the derivative is
taken with respect to (the variable at the denominator of the partial derivative) are
constant. Partial derivatives act like regular derivatives aside from that.
The force can be expressed by a vector expression:
⃗ = −(
𝑭
𝜕𝑈
𝜕𝑈
𝜕𝑈
⃗𝑈
𝑖̂ +
𝑗̂ +
𝑘̂) = −𝛁
𝜕𝑥
𝜕𝑦
𝜕𝑧
Module 9. Momentum
Summary:
•
Momentum/Linear Momentum (⃗𝒑)- a vector quantity, the product of its mass
and its velocity. Unit: kg*m/s
•
Impulse (𝑱)- a vector quantity, the product of the force and the time interval
during which it acts. Unit: N*s or kg*m/s
Changes in momentum depend on the time over which the net force acts
Changes in kinetic energy depend on the distance over which the net force
acts
Isolated System- when no external forces affect the system.
The total momentum is constant/conserved.
The total kinetic energy of the system is the same after the collision as before.
Both Kinetic Energy and Momentum are conserved in an elastic collision.
In a straight-line elastic collision of two bodies, the relative velocities before
and after the collision have the same magnitude but opposite sign
In an inelastic collision, the total kinetic energy after the collision is less than
before the collision.
Completely Inelastic Collision- A collision in which the bodies stick together.
Center of Mass- mass-weighted average position of particles
When an object or a collection of particles is acted on by external forces, the
center of mass moves as though all the mass were concentrated at that point
•
•
•
•
•
•
•
•
•
•
•
and it were acted on by a net external force equal to the sum of the external
forces on the system.
Formula:
9.1 Impulse and Momentum
• 𝑝 = 𝑚𝑣
• ∑𝐹 =
• 𝐽=
𝑑𝑝
𝑑𝑡
𝑡2
∫𝑡 ∑𝐹
1
𝑑𝑡
• 𝐽 = 𝐹𝑎𝑣 (𝑡2 − 𝑡1 )
• 𝐽 = 𝑝2 − 𝑝1
9.2 Conservation of Linear Momentum
• 𝑃 = 𝑝1 + 𝑝2 + ⋯ = ∑𝑝
9.3 Momentum Conservation and Collisions
•
Elastic Collisions
o
1
2
1
1
1
2
2
2
2
2
2
2
𝑚𝐴 𝑣𝐴1𝑥
+ 𝑚𝐵 𝑣𝐵1𝑥
= 𝑚𝐴 𝑣𝐴2𝑥
+ 𝑚𝐵 𝑣𝐵2𝑥
o 𝑚𝐴 𝑣𝐴1𝑥 + 𝑚𝐵 𝑣𝐵1𝑥 = 𝑚𝐴 𝑣𝐴2𝑥 + 𝑚𝐵 𝑣𝐵2𝑥
o 𝑣𝐵2𝑥 − 𝑣𝐴2𝑥 = −(𝑣𝐵1𝑥 − 𝑣𝐴1𝑥 )
• Perfectly Inelastic Collision
o 𝑚𝐴 𝑣𝐴1𝑥 + 𝑚𝐵 𝑣𝐵1𝑥 = (𝑚𝐴 + 𝑚𝐵 )𝑣2𝑥
𝑚𝐴
(𝑣𝐴1𝑥 )
o 𝑣2𝑥 =
𝑚𝐴 +𝑚𝐵
9.4 Center of Mass
• 𝑟𝑐𝑚 =
𝑚1 𝑟1 +𝑚2 𝑟2 +⋯
𝑚1 +𝑚2 +⋯
=
∑ 𝑖 𝑚 𝑖 𝑟𝑖
∑𝑖 𝑚 𝑖
• 𝑀𝑣𝑐𝑚 = 𝑚1 𝑣1 + 𝑚2 𝑣2 + ⋯ = 𝑃
Legend:
𝑝-momentum
𝐽- impulse
𝐹𝑎𝑣 - average force
Note: r may be any component (x,y, or z)
𝑟𝑐𝑚 -center of mass
𝑟- position/distance
𝑀- total mass
9.1 Momentum and Impulse
⃗ )- a vector quantity, the product of its mass and its
Momentum/Linear Momentum (𝒑
velocity. Unit: kg*m/s
𝑝 = 𝑚𝑣
Momentum is often shown in its components
𝑝𝑥 = 𝑚𝑣𝑥 ; 𝑝𝑦 = 𝑚𝑣𝑦 ; 𝑝𝑧 = 𝑚𝑣𝑧
Newton’s Second Law in terms of momentum
⃗ =
∑𝑭
⃗
𝑑𝒑
𝑑𝑡
Impulse (𝑱)- a vector quantity, the product of the force and the time interval during
which it acts. Unit: N*s or kg*m/s
𝐽 = ∑𝐹(𝑡2 − 𝑡1 )
If the net force is not constant, the impulse may be obtained from:
𝑡2
𝐽 = ∫ ∑𝐹 𝑑𝑡
𝑡1
or by defining an average net external force ⃗𝑭𝑎𝑣
𝑱 = ⃗𝑭𝑎𝑣 (𝑡2 − 𝑡1 )
Both impulse and momentum are vector quantities, their component forms are given
by:
𝐽𝑥 = (𝐹𝑎𝑣 )𝑥 (𝑡2 − 𝑡1 )
𝐽𝑦 = (𝐹𝑎𝑣 )𝑦 (𝑡2 − 𝑡1 )
Impulse-Momentum Theorem
⃗ 𝟐−𝒑
⃗ 𝟏 = ∆𝒑
⃗
𝑱=𝒑
In component forms:
𝐽𝑥 = 𝑝2𝑥 − 𝑝1𝑥
𝐽𝑦 = 𝑝2𝑦 − 𝑝1𝑦
Changes in momentum depend on the time over which the net force acts, but
changes in kinetic energy depend on the distance over which the net force acts
9.2 Conservation of Linear Momentum
Isolated System- when no external forces affect the system. The total momentum is
constant/conserved.
Total Momentum
⃗𝑷
⃗ = ⃗𝒑𝑨 + ⃗𝒑𝑩 + ⋯
In components
𝑃𝑥 = 𝑃1𝑥 + 𝑃2𝑥 + ⋯
𝑃𝑦 = 𝑃1𝑦 + 𝑃2𝑦 + ⋯
9.3 Momentum Conservation and Collisions
Elastic Collisions
•
•
•
The total kinetic energy of the system is the same after the collision as before.
Both Kinetic Energy and Momentum are conserved in an elastic collision.
Kinetic Energy
1
1
1
1
2
2
2
2
𝑚𝐴 𝑣𝐴1𝑥
+ 𝑚𝐵 𝑣𝐵1𝑥
= 𝑚𝐴 𝑣𝐴2𝑥
+ 𝑚𝐵 𝑣𝐵2𝑥
2
2
2
2
•
Momentum
𝑚𝐴 𝑣𝐴1𝑥 + 𝑚𝐵 𝑣𝐵1𝑥 = 𝑚𝐴 𝑣𝐴2𝑥 + 𝑚𝐵 𝑣𝐵2𝑥
•
In a straight-line elastic collision of two bodies, the relative velocities before
and after the collision have the same magnitude but opposite sign
𝑣𝐵2𝑥 − 𝑣𝐴2𝑥 = −(𝑣𝐵1𝑥 − 𝑣𝐴1𝑥 )
Inelastic Collisions
•
•
•
In an inelastic collision, the total kinetic energy after the collision is less than
before the collision.
Completely Inelastic Collision- A collision in which the bodies stick together.
If the external forces can be neglected, the total momentum is conserved.
𝑚𝐴 𝑣𝐴1𝑥 + 𝑚𝐵 𝑣𝐵1𝑥 = (𝑚𝐴 + 𝑚𝐵 )𝑣2𝑥
•
Special Case: B is initially at rest
𝑣2𝑥 =
𝑚𝐴
(𝑣 )
𝑚𝐴 + 𝑚𝐵 𝐴1𝑥
9.4 Center of Mass
Center of Mass- mass-weighted average position of particles
⃗𝒓𝑐𝑚 =
⃗ 1 + 𝑚2 ⃗𝒓2 + 𝑚3 𝒓
⃗ 3 + ⋯ ∑𝑖 𝑚𝑖 𝒓
⃗𝑖
𝑚1 𝒓
=
𝑚1 + 𝑚2 + 𝑚2 + ⋯
𝑚𝑖
Note: The components of vector use the formula, substitute x or y values to the
⃗ 𝑐𝑚 becomes 𝑥𝑐𝑚 or 𝑦𝑐𝑚 and 𝒓
⃗ 1 becomes 𝑥1 or 𝑦1 ).
vectors in the formula (e.g. 𝒓
Total Momentum
𝑀𝑣𝑐𝑚 = 𝑚1 𝑣1 + 𝑚2 𝑣2 + ⋯ = 𝑃
When an object or a collection of particles is acted on by external forces, the center of
mass moves as though all the mass were concentrated at that point and it were acted
on by a net external force equal to the sum of the external forces on the system.
References
Young, H. and Freedman, R. (15th Edition). University physics with Modern Physics.
New York: Addison-Wesley Publishing Company