Physics 121 Final Review PDF

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Physics 121 Final Review Chapter 1: Introduction, Measurement, Estimating: 1-1: The Nature of Science:  Observation: important first step toward scientific theory; requires imagination to tell what is important o Observations will tell if the prediction is accurate, and the cycle goes on  Theories: created to explain observations; will make predictions – well developed – can make testable predictions  Law: a theory that can be explained simply and which is widely applicable  Model: like an analogy – not intended to be a true picture – provides a familiar way of envisioning a quantity  How does a new theory get accepted? o Predictions agree better with data o Explains a greater range of phenomena 1-2: Physics and Its Relations to Other Fields:  Physics is needed in both architecture and engineering  Other fields that use physics, and make contributions to it: physiology, zoology, life sciences…  Communication between architects and engineers is essential if disaster is to be avoided 1-3: Models, Theories and Laws:  Models are very useful during the process of understanding phenomena. o A model creates mental pictures; care must be taken to understand the limits of the model and not take it too seriously  Theory: detailed and can give testable predictions  Law: brief description of how nature behaves in a broad set of circumstances  Principle: like a law, but applies to a narrower range of phenomena 1-4: Measurement and Uncertainty; Significant Figures:  No measurement is exact; there is always some uncertainty due to limited instrument accuracy and difficulty reading results  Estimated uncertainty is written with a ± sign; for example: 8.8 ± 0.1 cm  Percent uncertainty is the ratio of the uncertainty to the measure value, multiplied by 100  The number of significant figures is the number of reliably known digits in a number o It’s usually possible to tell the number of significant figures by the way the number is written:  23.21 cm has 4 sig figs  0.062 cm has 2 sig figs (the initial zeroes don’t count)  80 km is ambiguous – it could have 1 or 2 sig figs – if it has 3, it should be written 80.0 km  Terminology: most and Least significant digit:

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When multiplying, or dividing numbers, the result has as many sig figs as the number used in the calculation with the fewest sig figs o 11.3 cm x 6.8 cm = 77 cm When adding, or subtracting, the answer is no more accurate than the least accurate number (or sig digit) used o 123.67 + 8.9 = 132.57  132.6 (round answer – don’t truncate – to the correct sig fig)

 The # of sig figs are the # of digits used when writing a number in scientific notation 1-5: Units, Standards, and the SI System:  Length – Meter (m)  Time – Second (s)  Mass – Kilogram (kg)  Area: (m^2)  Volume: (m^3)  Speed: (m/s)  Force: (kg m/s^2)  Energy (Kg m^2/s^2)  Power (Kg m^2/m^3)  SI system: (MKS units) Meters, Kilograms, and Seconds 1-6: Converting Units:  Converting between metric units, for example from kg to g, is easy, as all it involves is powers of 10  Unit Cancellation 1-7: Order of Magnitude: Rapid Estimating:  A quick way to estimate a calculated quantity is to round off all numbers to one sig fig and then calculate – the result should at least be the right order of magnitude – this can be found by rounding it off to the nearest power of 10 1-8: Dimensions and Dimensional Analysis:  Dimensions: the base units that make up the quantity – generally written using square brackets o Ex: Speed = distance/time o Dimensions of speed: [L/T]  Quantities that are being added or subtracted must have the same dimensions  Dimensional analysis – useful for checking calculations Chapter 2: Describing Motion: Kinematics in One Dimension 2-1: Reference Frames and Displacement:  Any measurement of position, distance, or speed must be made with respect to a reference frame o If you are sitting on a train and someone walks down the aisle, their speed with respect to the train is a few miles per hour, at most – their speed with respect to the ground is much higher o We must define an “origin” to make measurements from – a positions (relative to origin) is noted as “x” and measured in meters  Distinction between distance and displacement o Displacement – how far the object is from its starting point, regardless of how it got there o Move to the left – negative displacement o Move to the right – positive displacement o Distance - traveled – measured from the actual path 2-2: Average Velocity:  Speed: How far an object travels in a given time

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Velocity includes directional information:

2-3: Instantaneous Velocity:  The instantaneous velocity is the average velocity, in the limit as the time interval becomes infinitesimally short

2-4: Acceleration:  Acceleration: the rate of change of velocity  

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Acceleration is a vector There is a difference between negative acceleration and deceleration: o Negative acceleration is acceleration in the negative direction as defined by the coordinate system o Deceleration occurs when the acceleration is opposite in direction to the velocity The instantaneous acceleration is the average acceleration, in the limit as the time interval become infinitesimally short

Vector Scaler (∆x) Displacement (m) (x) Distance (m) (v) Velocity (m/s) (v) Speed (m/s) (a) Acceleration (m/s^2) (a) Acceleration (m/s^2) 2-5: Motion at CONSTANT Acceleration:  The average velocity of an object during a time interval t is: o v = x/t o As the velocity is increasing at a constant rate:

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o Acceleration (assumed constant): o a = v/t Kinematic Equations:

2-6: Solving Problems: 1. Draw a diagram and choose a coordinate axes 2. Write down the know (given) quantities – then the unknown ones that you need to find in a table 3. Plan an approach to a solution – pick a suitable equation(s) 4. Check dimensions 5. Check units 2-7: Falling Objects:  Near the surface of the earth, all objects experience approximately the same acceleration due to gravity  In the absence of air resistance – all objects fall with the same acceleration  Acceleration due to gravity: 9.8 m/s^2 2-8: Graphical Analysis of Linear Motion:

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Slope of v-t graph: ACCELERATION Area under v-t graph: DISPLACEMENT

Chapter 3: Kinematics in Two Dimensions; Vectors 3-1: Vectors and Scalars:  Vector: has both MAGNITUDE and DIRECTION o Displacement, velocity, force, and momentum  Scalar: only has MAGNITUDE o Mass, time and temperature 3-2: Addition of Vectors:  For vectors in one dimension – simple addition and subtraction are all that is needed o Be careful about the signs (+ or -)  Resultant: the vector sum of the 2 or more vectors added or subtracted together  2 dimensions:

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Find displacement, using PYTHAGOREAN THEOREM Magnitude (displacement) is found by using Pythagorean Theorem Direction (Angle Ø) is found with Trig – measured as a positive angle

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USE “TAIL TO TIP” method to add vectors graphically To SUBTRACT vectors – you need to define the NEGATIVE of a vector  has the same magnitude but points in the opposite direction A vector V can be multiplied by a SCALAR c – the result is a vector cV that has the same DIRECTION but a MAGNITUDE of cV – if c is NEGATIVE – the resultant vector points in the opposite direction Vectors can be broken down into x/y components

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3-4: Adding Vectors by Components:  The vector sum V can be expressed in terms of unit vector i & j

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Adding Vector: o Draw a Diagram; add the vectors graphically o Choose x & y axes o Resolve each vector in x & y components o Calculate each component using sines and cosines o Add the components in each direction o To find the length and direction of the vector use:

3-5: Projectile Motion:  Projectile: an object moving in two dimensions under the influence of Earth’s gravity; its path is a parabola  It can be understood by analyzing the horizontal and vertical motions separately

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The SPEED in the x-direction is constant; in the y-direction the object moves w/constant ACCELERATION g If an object is LAUNCHED at an initial angle of θi with the horizontal, the analysis is similar except that the initial velocity has a VERTICAL component

3-6: Solving Problems Involving Projectile Motion:  Projectile motion is motion with constant acceleration in two dimensions, where the acceleration is g and is down

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Draw a diagram Choose an origin and a coordinate system Decide on the time interval  same in both directions and includes only the time the object is moving with constant acceleration g  Examine x and y motions separately  List known and unknown quantities  remember Vx never changes, and that Vy = 0 at the highest point  Apply equation  Only 45º gives a unique range – with no air resistance 3-8: Relative Velocity:  Consider relative speed in one dimension; it is similar in two dimensions except that we must add and subtract velocities as VECTORS  Each velocity is labeled first with the object and second with the reference frame in which it has this velocity – Vws is the velocity of the water in the shore frame, Vbs is the velocity of the boat in the shore frame, and Vbw is the velocity of the boat in the water frame

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Chapter 4: Dynamics: Newton’s Laws of Motion 4-1: Force:  Force: is a push or pull – an object at rest needs a force to get it moving – a moving object needs a force to changes its velocity  The magnitude of a force can be measured using a spring scale 4-2: Newton’s First Law of Motion:  Newton’s first law of Motion: often called law of inertia  Every object continues in its state of rest, or of uniform velocity in a straight line, as long as no net force acts on it  Inertial reference frames: one in which Newton’s first law is valid – excludes rotating and accelerating frames 4-3: Mass vs Weight:  Mass: the measure of inertia of an object – measure in kg  Mass is not weight (weight = mg) o If you go to the moon, whose gravitational acceleration is about 1/6 g, you will weigh much less, your mass will be the same 4-4: Newton’s Second Law of Motion:  Deals with relation between acceleration, force, & mass 

Force is a vector so is true along each coordinate axis o Measured in newton (N) = kg*m/s^2 4-5: Newton’s Third Law of Motion:  

Any time a force is exerted on an object, that force is caused by another object Newton’s Third Law: whenever one object exerts a force on a second object, the second exerts an equal force in the opposite direction on the first (Fn)  Reaction force – what propels the rocket 4-6: Weight – the Force of Gravity; and the Normal Force:  Weight: the force exerted on an object by gravity – when gravitational force is nearly constant – the weight is: Fg = mg  An object at rest must have no net force on it – if it were sitting on a table, the force of gravity is still there – and normal force from the table (perpendicular to a surface)  Normal Force – exactly as large as needed to balance the force from the object (of the required force gets too big – something breaks) 4-7: Solving Problems with Newton’s Laws – FBD  Draw a sketch  For one object, draw a FBD – showing all the forces acting on the object – label each force – if there are multiple objects – draw a FBD for each one  Resolve vectors into components  Apply Newton’s second law to each component  Solve

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Typical forces:

o Normal Force o Gravity o Tension: in ropes & cords, etc. o Friction: a resistive force  Equilibrium: when the sum of the forces (net force) of system = 0  A system can be in equilibrium in one dimension, but not the other 4-8: Application Involving Friction, Inclines:  On a microscopic scale, most surfaces are rough. The exact details are not yet known, the force can be modeled in a simply way    

For kinetic – sliding – friction: is the coefficient of kinetic friction – different for every pair of surfaces is the coefficient for static friction – describes how tightly two nonmoving surfaces grip one another Static friction: the frictional force between two surfaces that are not moving along each other – keeps objects on inclines from sliding – keeps objects from moving when a force is first applied

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The static frictional force increases as the applied force increases, until it reaches its maximum – then the object starts to move, and the kinetic frictional force takes over

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An object sliding down an incline has 3 forces acting on it: the normal force, gravity, and the frictional force o The normal force is always perpendicular to the surface o The friction force is parallel to it o The gravitational force points down o We align the x- axis with the plane of the incline, and the y-axis perpendicular to the incline

Chapter 5: Circular Motion; Gravitation:

5-1: Kinematics of Uniform Circular Motion:  Uniform circular motion: motion in a circle of constant radius at ….  Instantaneous velocity: is always …  To determine the direction of acceleration, examine the change in velocity (∆v) over a very small region of the circular path.

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This acceleration is called centripetal or radial, acceleration – always points toward the center of the circle 5-2: Dynamics in Uniform Circular Motion:  For an object to be in uniform circular motion, there must be a net force acting on it  Net force must be inward – think about a ball on a string 

There is no centrifugal force pointing outward; what happens is that the natural tendency of the object to move in a straight line must be overcome – if the centripetal force vanishes, the object flies off tangent to the circle  When a car goes around a curve, there must be a net force towards the center of the circle – if road is flat – that force is supplied by friction  If the frictional force is insufficient – the car will tend to move more nearly in a straight line  As long as the tires don’t slip – the friction is static – if the tires start to slip – the friction is kinetic o The kinetic frictional force is smaller than the static o The static frictional force can point towards the center of the circle – but the kinetic frictional force opposes the direction of motion, making is very difficult to regain control of the car and continue the curve 5-3: Highway Curves, Banked and Unbanked  Banking the curve can help keep cars from skidding – for every banked curve there is one speed where the entire centripetal force is supplied by the horizontal component of the normal force – no frictional force is required

5-4: Nonuniform Circular Motion:  If an object is moving in a circular path but at varying speeds, it must have a tangential component to its acceleration as well as the radial one  This concept can be used for an object moving along any curved path, as small segment of the path will be approximately circular 5-5: Centrifugation:  A centrifuge works by spinning very fast – this means there must be a very large centripetal force 5-6: Newton’s Law of Universal Gravitation:  If the force of gravity is being exerted on objects on Earth, what is the origin of that force?  Force keep the Moon in its orbit  The gravitational force on you is one-half of the Third Law pair – the Earth exerts downward force on you, and you exert an upward force on the Earth  Earth and Moon exert equal and opposite forces  The gravitational force must be proportional to both masses  The gravitational force must decrease as the inverse of the square of the distance between the masses  Newton’s Las of Gravitation:

5-7: How do you mass the Earth:  We can relate the gravitational constant to the local acceleration of gravity

5-8: Satellites and “Weightlessness”  The tangential speed must be high enough so that the satellite does not return to Earth, but not so high that it “escapes: Earth’s gravity altogether  The satellite is kept in orbit by its speed –it is continually falling  Objects in orbit are said to experience weightlessness  The satellite and all its contents are in free fall – there is no normal force  apparent weightlessness o Gravitational force still exists Chapter 6: Work and Energy: 6-1: Work Done by a Constant Force:  Work is an expression of energy used – work is SCALAR  The work done by a constant force is defined as the displacement moved multiplied by the component of the force in the direction of the displacement

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SI units of work: Joules (1J = 1N*m) Solving work Problems: o Draw FBD: identify forces o Choose coordinate system o Apply Newton’s laws to determine any unknown forces o Find the work done by a specific force o To find net work – find the net force and then find the work it does or find the work done by each force and add  Work done by force that oppose the direction of motion (friction) will be negative  Centripetal forces do no work, as they are always perpendicular to the direction of motion 6-2: Work done by varying force:  For a force that varies – the work can be approximated by dividing the distance up into small pieces, finding the work done during each and then adding them – the work is also the AREA under the force vs. distance/time curve 6-3: Kinetic Energy, and the Work-Energy Principle:  Energy was traditionally defined as the ability to do work – all forces can do work  Energy of motion is called KE – it’s scalar  KE = (1/2) mv^2 o Wnet = ∆KE  If the net work is positive  KE increases  If the net work is negative  KE decreases  Work and KE are measuring JOULES 6-4: Potential Energy:  PE: depends on surrounding o A wound-up spring o A stretched elastic band o An object at some height above the ground  In raising a mass m to a height h at constant speed, the work done by the external force is: o Wext = Fext*h*cos 0 = mgh o PEg = mg(y2-y1) = mg∆y  This PE can become KE if the object is dropped  PE is a property of a system as a whole – not just the object – depends on external forces (gravity)  PE can be stored in springs when it is compressed o Fs = -kx o X = the spring displacement (from equilibrium) o K = the spring constant o PEe = (1/2)kx^2  Of the compressed or stretched spring – measured from its equilibrium position 6-4: Potential Energy:  The force increases as the spring is stretched or compressed further 6-5: Conservative and Nonconservative Forces:  If friction is present – the work done depends not only on the starting and ending points – but also on the path taken – Nonconservative force (friction)

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Conservative force is path independent (gravity) Distinguish between work done by conservative forces and the work done by Nonconservative forces Work done by Nonconservative forces is equal to the total change in KE and PE o Wnc = ∆KE + ∆PE o Total mechanical energy o If Wnc = 0 – there is no change in the total mechanical energy o If there are no Nonconservative forces – the sum of the changes in the KE and in the PE, is ZERO – the KE and PE changes are equal but opposite in sign o Total mechanical energy: E = KE + PE o E1 = E2

o Total ME when dropping an object 6-7: Problem Solving using conservation of Mechanical Energy:  If there is no friction  the speed will depend only on its height compared to its starting height  For an elastic force – conservation of energy:

6-8: Other forms of energy:  Work is done when energy is transferred from on object to another  Energy is conserved (E1 = E2) – as a whole – when only conservative forces are acting 6-9: Energy conservation:  Heat = Wnc  If there is a Nonconservative force such as friction – the kinetic and potential energies become heat  Problem Solving: o Draw picture o Determine system where energy will be conserved o Decide on initial and final positions o Choose a logical reference frame o Apply conservation of energy – Wnc (most likely friction) is added on left ...