Mechanics: Work, Energy, Momentum, Kinematics of Rotational Motion

Mechanics: Work, Energy, Momentum, Kinematics of Rotational Motion
Jacque Lynn F Gabayno, Ph.D. Lecture Notes
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Work as defined in Physics

r r W =F s

Force× Displacement = Force × Displacement
*SI Units: 1 N.m = 1 Joules (i.e. same as the unit of energy)
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Recall “Dot Product”
The dot product allows us to multiply two vectors to get something that is SCALAR.

r A r A
For a constant force:

r r r B = A B cos ! r B = Ax Bx + Ay By + Az Bz

Only those along the direction of motion contribute to the total work done on an object.
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With position-dependent forces
F
Vector sum of all forces acting on the body Area under the curve

xf

W =

! xi r r F x

xi

xf

x

The work done by a variable force is equivalent to the area under the force-distance curve along the path of the object.
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Work “Polarity”: Positive or Negative (could even be Zero)
If force applied is … parallel to displacement antiparallel to displacement perpendicular to displacement Work done is … positive negative zero
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Work: Example
Fapp FN
Direction of motion

Friction, Ffr FW

Negative work reduces the kinetic energy of a system.

Friction ALWAYS does negative work Gravity does zero work ONLY if the motion is parallel to the/along a horizontal surface
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Work-Kinetic energy relation

Wtot = !KE = K f " K i
Due to APPLIED FORCES
(e.g. elastic force (following Hooke’s law), friction, gravity, tension, normal force, etc.)

For a particle in Linear motion

1 2 K = mv 2
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POWER* = how FAST work gets done
Average Power Instantaneous Power

r r !W P = =F v !t


r r dW P= =F v dt

SI Units:  1 watts = Joules/sec  1 horsepower (hp) = 745 W

*Would be nice if this also applies in government :p

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Energy has two faces
Kinetic energy:  anything associated with motion (translational, rotational) Potential energy:  something that is stored for later use  Gravitational potential energy  Elastic potential energy  Chemical potential energy  Nuclear potential energy

U grav = mgy 1 2 U el = kx 2
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Work and Potential energy relation (2nd Party)

Wtot = !"U = U i ! U f
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Their Love Story

!K + !U = 0
Any decrease in PE results to an increase in KE

--> Conservation of Mechanical Energy --> valid only if NO third party (e.g. Friction) is involved
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But if friction is present…

!K + !U + !U other = 0
It’s the TOTAL energy that is conserved
In an isolated system, no energy is created nor destroyed: says the Universe

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Conservative vs Non-conservative Forces
Conservative force if the work it does on a particle that completes a round trip is zero; otherwise the force is nonconservative.
Gravity and elastic force are conservative: can do both positive and negative work

Also conservative if the work done by it on a particle that moves between two points is the same for all paths connecting these points; otherwise the force is non-conservative
Frictional force is non-conservative: only does negative work

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Force and Potential Energy
A little dose of Calculus

1D

3D
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From Potential Energy Diagrams

dU =0 dx
U (x)

Establishes equilibrium

U (x)

stable equilibrium (U is minimum)

unstable equilibrium (U is maximum)

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r r p ! mv

Linear Momentum
For a single particle

Whose components are

px = mvx p y = mv y pz = mvz


SI Units: kg.m/s
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x, y, and z components are independent

Fundamental relation to Newton’s law

The rate of change of momentum (either linear or angular) is proportional to the net force acting on the system.
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The CHANGE in momentum due to an external force is called impulse

For a constant force

For a non-constant force

r r Fav !t = !p



SI Unit: kg.m/s
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If there are no external forces (or if their sum is zero)…
The total momentum of a system is conserved.

r r P = Pf i conserved. Px = constant Py = constant Pz = constant

 Is ALWAYS TRUE even if the total mechanical energy is not
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Momentum to describe Inelastic collision (real-life collision)
After inelastic collision, the kinetic energy is not conserved. It is ALWAYS LESS than the initial kinetic energy. Some of it is lost to • Heat (due to friction) • Deformation (such as bending of metals)
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Perfectly Inelastic collision

(no friction)

By COM: MVo = (m + M )Vf
M 2m Vf = V0 = V0 m+M 2m + m 2 Vf = V0 21 3

Momentum to describe Elastic collision
Just like momentum, the kinetic energy is always conserved after perfectly elastic collision. Their relative velocities are:

v1i + v1 f = v2 f + v2i
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A particle no more
The effective (or average) position of a complex object (such as yourself) is at the CENTER OF MASS. It doesn’t change as long as there are no external forces.

For a collection of particles

For a continuous system

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“You complete me…”
Linear Displacement Velocity Acceleration Momentum -Particle -Collection of particles - Rigid body Kinetic Energy Force Work Power s v atan or arad p = mv pcm = Mvcm ½ mv2 F F·s F·v

(Rotation)
Angular Δθ = rs ω = rv α = atan/r L = r x p (park) L = I ω (park) ½ I ω2 τ = r x F (park) τ · Δθ (park) τ · ω (park)
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These too…(Kinematics)
Linear/Translation Angular/Rotation

a x = constant 1 2 x = xo + vox t + a x t 2 v = vox + a x t

! x = constant
1 2 # = # o + "oz t + ! z t 2 " = "oz + ! z t

v = v + 2a x (x ! xo ) 1 x ! xo = (vx + vox )t 2

2 x

2 ox

! = ! + 2# z (" $ " o ) 1 " $ " o = (! z + !oz )t 2
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2 z

2 oz

Right Hand(ed) Rules axis of rotation

α ω

(ω and α are along same direction)

α ω

(ω and α are along opposite direction)

x y

x y

Direction of rotation Speeding up

Direction of rotation Slowing down
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Moment of Inertia

The GREATER the I, the harder it is to start a body to rotate (when initially at rest) or stop rotating (when initially moving).
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Linear and Angular Mechanics Concepts
Newton’s 2nd Law Work-Kinetic Energy Theorem Linear F=ma Wtot = ΔKE Angular τ=Iα Wtot = ΔKEtran + Δ KErot

Conservation Rules (Isolated Systems) 1. Energy Ui + ½ mvi2 + ½ Iωi2 = Uf + ½ mvf2 + ½ Iωf2 pi = pf : (Fext = 0) Li = Lf : (τext = 0)

2. Linear Momentum 3. Angular momentum

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To shift the axis of rotation… d I O = I CM + Md
Parallel to the original axis

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From me to you…
To see how these concepts apply to actual mechanics problems, you have to understand the examples, homeworks, seatworks, and group problems. AGAIN, I don’t recommend that you memorize formulas.

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Next: Conditions for Equilibrium, Oscillation, Elasticity, Fluid Mechanics
Jacque Lynn F Gabayno, Ph.D. Lecture Notes
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