Physics 111 Semester Review

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Physics 111Semester Review

• Kinematics
• Dynamics
• Work, Kinetic Energy, Potential Energy
• Momentum, Collisions
• Gravity
• Oscillations
• Waves
• Fluids/Kinetic Theory
• Thermodynamics

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Kinematics, Constant Acceleration

• Position x x0 v0,x t (1/2) ax t2
• The x-coordinate at time t is equal to the
  initial x-coordinate x0 plus the initial velocity
  v0,x times t plus one-half the acceleration ax
  times t -squared.
• Velocity vx v0,x ax t
• Velocity equals initial velocity plus
  acceleration times time
• Solve velocity equation for t, plug into position
  equation
• (x- x0) ax (1/2) (vx 2- v0,x 2)
• An equation of position and velocity, without
  explicit reference to time.
• Measure acceleration from a velocity vs time
  graph
• a (v2-v1) / (t2-t1)
• Acceleration equals change in velocity divided by
  time to achieve the change.

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Vectors trigonometry
C
A

• Vector components add separately
• A B C
• Ax Bx Cx
• Ay By Cy
• If you put the origin of a coordinate system at
  the tail of vector A, and figure out the angle q
  between the x-axis and A, then
• Ax A cos q
• Ay A sin q

B
x
A
Ay
q
Ax
x
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Motion in 2 (or more) dimensions

• If the motion is in two dimensions,
• each coordinate can be analysed separately.
• Just replace x ??y in equations
• Example Trajectory motion, free fall
• Launch projectile at an angle q from horizontal
• with initial speed v0
• v0,x v0 cos(q) v0,y v0 sin(q)
• ax 0 ay -g
• x x0 v0,x t y y0 v0,y t - (1/2) g t2
• 0 (vx 2- v0,x 2) -(y- y0) g (1/2) (vy 2-
  v0,y 2)

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Dynamics

• Inertia resistance of object to change in state
  of motion.
• Mass measure of inertia
• Forces act to change the state of motion of an
  object
• Net Force Sum of all forces acting on a mass m
• This is statement that forces simply add
• Net Force mass times acceleration
• This is Newtons 2nd law, linking the force with
  motion
• Notice that individual forces do not separately
  acceleration, only the sum of all forces
• Fnet m a
• Fnet 0 if and only if acceleration is zero
• If acceleration is zero, then velocity does not
  change.

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Examples of Forces

• Gravity (near surface) F mg (down)
• Gravity acting on a mass m a distance r from
  center of mass M
• Earth radius RE, Mass ME, mass m at height h
  above surface, h ltlt RE,
• Tension Pulls object only in direction of
  string
• Spring force F - k (x-x0)
• x0 equilibrium position (often x0 0).

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Forces of contact

• The force at the contact between two objects
• separated into its components parallel and
• perpendicular to the surface
• Force parallel Friction force
• Force Perpendicular normal force N
• Normal force acting on m points into m
• (this is the no glue hypothesis).
• Normal force has whatever magnitude necessary to
  keep v_perp0
• (this is the no walking through walls
  hypothesis).
• If a block slides on an incline, or a cart rolls
  on a track, v? 0
• Even for a ball bouncing on floor, v ? 0 at
  moment of contact (even though v ? ? 0)
• Normal force is NOT mg. Sometimes N has same
  value as Gravitymg.

Ff
N
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Apparent Weight Weightnessless

• If you stand on a scale, the force of the scale
  pushing up on you (the force N) is your APPARENT
  WEIGHT.
• This force is bigger than mg (the force of
  gravity acting on you) if you are standing in an
  elevator accelerating upwards
• N - mg ma
• This force is smaller than mg if you are standing
  in an elevator accelerating downwards.
• This force is zero if you are an astronaut in the
  space-station in free-fall.
• Weightnessless does not mean force of gravity 0.

a
mg
N
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Static Friction force

• Static Friction,
• Mass m is not moving relative to surface of
  contact
• FS lt mSN
• The force of static friction takes on whatever
  magnitude or direction (but parallel to surface)
  necessary to keep v?? 0 (component of velocity
  parallel to surface).
• Note if v?? 0 and constant, then a?? 0.
• But FS cannot exceed the numerical value mSN.

Fs
N
mg
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Kinetic Friction
v

• Magnitude of the force of kinetic friction is
  fixed
• FK mKN
• Direction of force of kinetic friction on mass m
  opposes the slipping of mass m on the surface.
• Acceleration can be positive, negative, or zero.

FK
N
mg
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Free Body Diagramswww.physics.odu.edu/hyde/Teach
ing/Lectures/FreeBodyDiagrams.html

• Draw a sketch
• Draw a separate sketch of just the mass m.
• Draw all forces acting on mass m
• Give each force a unique label N, mg, fs
• Do not give the same force more than one label.
• Draw all forces again with all tails at the
  origin of a coordinate system.
• Evaluate x- and y-components of forces
• Nx0 NxN, Fk,x -FK, Fk,y 0
• Wx mgcos(270-q) Wymg sin(270º-q)
• Identify all constraints
• FK mK N
• Action-Reaction partners.
• Apply Newton 2nd Law Sum of all forces acting
  on mass m equals m times acceleration a of mass
  m.
• Fnet ma

v
q
FK
N
mg
N
x
FK
mg
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Work, Kinetic Energy, Mechanical Energy,
Potential Energy

• In kinematics of freefall, we already got a hint
  of the role of energy
• -(y- y0) g (1/2) (vy 2- v0,y 2) 0 (vx 2-
  v0,x 2)
• Gravity is a conservative force. Near surface of
  earth, gravitational potential energy mgy.
• Free Fall, Mechanical energy is conserved E K
  U constant, or DK DU 0, or -DU DK
• -DU -(mgy mg y0) DK (1/2) m v 2- (1/2)m v2

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Work
d
q
F

• A force F acts on a mass m while the force moves
  through a displacement d (there may be other
  forces also).
• The Work W done by force F is W F d cosq
• q is the angle between the force and the
  displacement.
• The total Work WTotal is just the work done by
  the net force. This is also the sum of the works
  done by each force.
• WTotal Fnet d cosq
• q is the angle between net force Fnet and
  displacement d
• WTotal F1 d cosq1 F2 d cosq2
• q1 is the angle between force F1 and
  displacement d.
• Example A frictionless stationary track does no
  work on a sliding/rolling object The normal
  force is perpendicular to the motion (by
  definition).
• Use work to analyze motion on track without every
  worrying about that confusing Normal-force!!

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Work Energy Theorem

• Total Work done on mass m equals change in
  kinetic energy of m.
• Wtotal Kf Ki DK
• K (1/2) m v2

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Conservative Forces

• A conservative force is a force that depends only
  on position
• Gravity
• Spring force F -kx
• Electrostatics (next semester).
• A non-conservative force depends also upon the
  present (or past) state of motion.
• Friction

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Potential Energy

• A conservative force depends only on position.
• A particle therefore has the ability to change
  its state of motion, just by virtue of its
  position. This is the potential to do work, or
  potential energy.
• If the conservative force F does work W on a
  particle of mass m (other forces may also be
  present), define the potential energy U
  associated with the conservative force F as
  follows
• W -DU - (Uf -Ui)
• This only defines changes in potential energy,
  the location x0 such that U(x0)0 is yours to
  choose at will.

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Examples of Potential Energy

• Ideal Hookes Law Force
• U (1/2) k x2
• Gravity (a mass m a distance r from center of
  mass M). We choose U?0 as r??.
• Gravity a distance h above surface of Earth
  (radius RE, mass ME)

For motion near earth surface, use U(RE h)-U(RE)
mgh
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Mechanical Energy E

• E K U kinetic potential energy
• WNC Work done on mass m by non-conservative
  forces.
• WNC Ef Ei DE
• If WNC 0, then mechanical energy is conserved
• DE 0 ? Ef Ei

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Thermal Energy

• Total energy includes internal Thermal Energy Uth
• The work done by nonconservative forces is
  converted into heat (some goes into system, some
  into surroundings).
• Change in Thermal energy Uth Heat into system
  minus work done by system
• D Uth Q - W

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Momentum Collisions

• Momentum p mv
• Fnet m a or Fnet rate of change of momentum
• If there are no external forces, then momentum is
  conserved (e.g. collisions)
• p1,i p2,i p1,f p2,f
• Good way to analyze motion without worrying about
  details of forces
• Elastic collision, Total kinetic energy is
  conserved
• Inelastic collision, Total momentum conserved,
  but kinetic energy decreases (splatheat) or
  increases (explosion).

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Oscillations

• Mass m on a spring with ideal Hookes law spring
  constant k
• F - k x
• E (1/2) m v2 (1/2) k x2 constant.
• Motion is periodic with period 2p/w
• w k/m1/2
• X(t) A cosw(t-t0)

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Waves

• Waves on string, velocity v F/s1/2
• F tension in string
• s M/L mass per unit length
• Wavelength, frequency, velocity
• v f l

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Fluids, Kinetic Theory, Thermodynamics