Physics 210 Final Exam Study Guide
Physics 210 Final Exam Study Guide Phys 210
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This 87 page Study Guide was uploaded by Maria Dion on Saturday December 5, 2015. The Study Guide belongs to Phys 210 at Northern Illinois University taught by Michael fortner in Fall 2015. Since its upload, it has received 298 views. For similar materials see General physics I in Liberal Arts at Northern Illinois University.
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Physics 210 Final Exam Study Guide Unit 1 o Unit Systems Systems set up fundamental units British system foot, pound, second Metric system meter, kilogram, second o Time The unit of time originally was based on the day and the year. The second was 1/60 * 1/60 *1/24 of a day. In the 20th century the second was measured based on the timing of atoms. We now know that the day is getting longer and “leap seconds” are added every few years. The SI unit of time is the second (s) o Length The oldest standards of length were based on the human body. The metric system defined the meter in terms of the Earth: 1/10,000,000 from the pole to the equator. The meter is now defined in terms of the second and speed of light. The SI unit of length is the meter (m) o Mass Standard weights have been maintained for centuries. Weight and mass were thought to be the same. Now a standard 1 kilogram mass is kept in Paris. The SI unit of mass is the kilogram (kg) o SI Prefixes Prefixes on units are used to represent powers of ten. Prefixes denote powers of ten from 18 to +18 in steps of three. Example: a kilometer is 103 meters or 1000 meters. Most Common micro (m) 106 milli (m) 103 kilo (k) 103 mega (M) 106 Common, but the power is not a factor of three. centi (c ) 102 deci (d ) 101 o Other Units There are other fundamental units in SI. See NIST. ampere (A) kelvin (K) mole (mol) candela (cd) Derived units are built from the fundamental units. area (m2) volume (m3) velocity (m/s) acceleration (m/s2) force (kg m/s2) or (N) energy (kg m2/s2) or (J) o Special Units Earth to Sun 149,597,871 km = 1.5 x 1011 m 1 Astronomical Unit (1 AU) (1.5 x 1011 m) / (3.0 x 108 m/s) / (60 s/min) = 8.3 lightminutes o Units and Types Units are meters, seconds, feet, tons, etc. Types of units are length, mass, force, volume, etc. The type of unit of a value is called the dimension. A value in square meters has dimensions of an area. A value in kilometers per hour has dimensions of a velocity. o Matching Units Conversion between units must be of the same type. Length conversion: 1 in = 2.54 cm Time conversion: 1 hr = 3.6 x 10 s No conversion between different types of units. 1 in is not equivalent to some seconds o Conversion Factors A value is converted by applying the ratio of the conversion factors. How many inches in 50. cm? 50. cm (1 in / 2.54 cm) = (50. / 2.54) in = 20. in Many conversion factors use scientific notation. How many seconds in a year? 1 yr (365 d/yr) (24 hr/d) (3.6 x 103 s/hr) = 31500 x 10 s = 3.15 x 10 s o Powers of Units It is useful to convert the dimensions of units into fundamental dimensions. Length (L) Time (T) Mass (M) Units can be raised to a power, and so can the fundamental dimensions. Area (L2) Volume (L3) Force (M L / T2) o Missing Units The energy in a compressed spring is given by U = ½ kx2. U is the energy in kg m2/s2, and x is the length in m. What are the correct units for k? Use dimensional analysis: Substitute units for dimensions: k has units of kg/s2 Dimensional Expressions The speed of waves in shallow water depends only on the acceleration of gravity g, with dimensions L/T2, and on the water depth h. o Base Quantities Acceleration g dimensions: L/T2 length/time2 example: m/s2 Height h dimensions: L length example cm Speed v dimensions: L/T length/time example km/h o Limitations Dimensional analysis only checks the units. Numeric factors have no units and can’t be tested o Scientific Notation In physics numbers can be very large and very small. Scientific notation uses powers of 10 to represent decimal places. 5 Positive powers for large numbers: 456000 = 4.56 x 10 Negative powers for small numbers: 0.00753 = 7.53 x 10 3 o Order of Magnitude 0 You 5’9” = 1.75 m ? 10 m Lecture Hall 14 m ? 10 m Faraday West 80 m ? 10 m 3 NIU 2000 m ? 10 m Each of these lengths is different by about one order of magnitude o Accuracy The smallest unit on a measuring device sets the accuracy. In general, a measurement is only as accurate as the smallest unit. Significant figures are a guide to the accuracy of a measurement. o Significant Figures Any value is expressed in some number of digits. The number of digits (without left side zeroes) is the number of significant figures. With no decimal point, skip right side zeroes. Examples: 38 2 digits, 2 significant figures 5.06 3 digits, 3 significant figures 0.0041 5 digits, 2 significant figures 7,000. 4 digits, 4 significant figures 2,000 4 digits, 1 significant figure o Using Significant Figures Add or Subtract Keep the significant figures to decimal place of the least accurate value, rounding as needed. 4.361 + 14.2 = 18.6 12000 + 364 = 12000 Multiply or Divide Keep the same number of significant figures as the value with the fewest, rounding as needed. 4.361 x 14.2 = 61.9 12000 x 364 = 4.4 x 106 o Absolute Uncertainty Measure 50.0 cm. There are three significant figures. The smallest figure suggests an accuracy of 0.1 cm. This is also equal to 1 mm. The absolute uncertainty has the same type of units as the measurement. o Percent Uncertainty Measure 50.0 cm. Compare 0.1 cm to 50.0 cm. The ratio is 0.1/50.0 = 0.002. Multiply by 100 % to get 0.2 %. The percent uncertainty has no units, and is either a pure number or a percent. o Rounding The simplest estimation technique is to round. This works very well on formulas where all the values can be reduced to one significant figure. o Order of Magnitude Rounding Rounding to a power of ten is the crudest form of rounding. Order of magnitude estimates are easy to compare since they are all only powers of ten. For a comparison to work, the units need to be the same (meters and meters, not km). o Using Geometry Geometrical shapes can often be used to approximate real shapes. The standard formulas from geometry can be used to make an estimate. Shapes can be 2dimensional (triangle, circle) Or 3dimensional (box, sphere). o How Big? Assume the density of a rock is three times that of water. How many centimeters across is a one metric ton (1000 kg) rock? The rock has a density of 3 g/cm3 so the volume is 106 g / (3 g/cm3) = 3.3 x 105 cm3. Estimate that the rock is a sphere, V = (4/3) p r3 d = 2r = 2 (3V/4 p)1/3 d = 85.7 cm » 90 cm o Apparent Shift A moving observer sees fixed objects move. Near objects appear to move more than far objects. Telephone poles whip by faster than distant trees. The effect is due to the change in observation point, and is used by our eyes for depth perception. o Observing Parallax Observe an object against the background. Shift one seat left and observe again. Subtract to get the parallax shift. o How to Measure Measuring instruments are common. Ruler Clock Speedometer Thermometer Bathroom scale All instruments have a scale. Scale can be analog or digital Instruments can have multiple scales o Analog Scales Analog scales require interpolation and rounding Rounding when a value is taken at the nearest tick mark Interpolation when a value is estimated between two adjacent marks o Digital Scales Digital scales give a direct numeric value. Usually accurate to all digits in display. Many provide an extra digit for rounding. Calculators are not measuring devices. They will give more figures than are significant. Round to the correct number of significant figures o Data Tables Data is often placed in tables. Columns should be labeled with measurement and units Values should have the correct significant figures o Graphs Both a recording tool and measuring device Keep track of measurements as they are recorded Estimate measurements from data on the graph Graphs have two scales Unit 2 o Scalar Quantities Any measurement that consists of a single number is a scalar. 72 °F 500 milliliters 2.54 centimeters 6 hours, 24 minutes (consider this as 6.4 h) Most measured quantities consist of only a single magnitude. o Vector Quantities A measurement that requires more than one value to describe it is a vector. 10 km to the northeast At 41.9° N latitude and 88.7° W longitude 15 pounds of force directed down These quantities can be thought of as carrying the value from a starting point to a destination. The word vector means carrier. o Vector Diagram One representation of a vector is an arrow. The tail shows the start of the vector. The tip points in the direction. The length of the arrow shows the magnitude. o Vector Notation A vector variable is represented by a small arrow over the top of the variable. Some texts use boldface for vectors, but that can be hard to distinguish on some backgrounds. Our text uses both boldface and an arrow. The magnitude of a vector is a scalar. It is represented as the absolute value of the vector, or just the variable without the vector symbol. o Vectors in Equations Vector variables can be used in equations Vector variables are different from scalar variables o Graphical Addition The two vectors can be added graphically. The tail of the second vector is placed at the tip of the first. The length and directions are kept the same. The result is the total vector. Its magnitude can be measured on the graph. o Parallelogram Force vectors act on a common object at a single point. If two vectors are added from a common origin one can be shifted to make a parallelogram. This is the same as putting the tail to the tip. o Commutative Property Vectors can be shifted as long as they don’t change direction and magnitude. Vectors can be added in reverse order and get the same result. o Parallel and Antiparallel Vectors that point in the same direction are parallel Vectors that point in opposite directions are antiparallel. o Cancellation What happens if we add two antiparallel vectors of equal magnitude? The vector sum is a zero length vector. The vectors cancel out. o Coordinates Vectors can be described in terms of coordinates. 6.0 km east and 3.4 km south 1 N forward, 2 N left, 2 N up Coordinates are associated with axes in a graph. o Use of Angles Find the components of vector of magnitude 2.0 N at 60° up from the xaxis. Use trigonometry to convert vectors into components. x = r cos q y = r sin q This is called projection onto the axes. o Ordered Set The value of the vector in each coordinate can be grouped as a set. Each element of the set corresponds to one coordinate. 2dimensional 3dimensional The elements, called components, are scalars, not vectors. o Component Addition A vector equation is actually a set of equations. One equation for each component Components can be added like the vectors themselves o Vector Length Vector components can be used to determine the magnitude of a vector. The square of the length of the vector is the sum of the squares of the components. o Vector Direction Vector components can also be used to determine the direction of a vector. The tangent of the angle from the xaxis is the ratio of the ycomponent divided by the xcomponent. o Components to Angles Find the magnitude and angle of a vector with components x and y. o Alternate Axes Projection works on other choices for the coordinate axes. Other axes may make more sense for a particular physics problem. o Force Force is A push or pull on an object. A vector with magnitude and direction. Force is not Energy. Power. Momentum. Velocity. o Fundamental Forces Gravity is a fundamental force. It acts upon objects from a distance away from the source (such as the Earth). There are two other fundamental forces. o Contact Forces Many forces are due to contact between objects. Kick a ball Push with a bulldozer Tug from a rope Friction due to the ground The actual force is electricity, but the atoms are so small we can treat the forces as coming from contact by larger objects. o Newton’s Laws Ancient scientists looked to the natural properties of objects. Motion was a result of the object’s properties. Newton defined motion based on forces acting from outside an object. Motion was the result of external forces. Three laws were used to define the behavior of forces on objects. First Law: Law of Inertia An object continues at rest, or in uniform motion in a straight line, unless a force is imposed on it. This describes constant velocity, including zero. No change means no force, and vice versa. o Zero Net Force An object at rest with no net force is in static equilibrium. The net force is due to the sum of forces acting on the object. The forces are vectors o Vector Forces With no motion, forces must sum to zero. Free body diagram Identify forces on the sign. C is the force on the chain B is the force on the beam W is the weight Vector sum is zero. o Force Components To find the values, use components Find the vertical components for the force on the chain. Cy = C sinq Wy = 210 N 0 = Cy + Wy = C sinq + Wy C = Wy / sinq = 370 N Use horizontal components for the force on the beam. 0 = Bx + Cx = Bx + (C cosq) Bx = C cosq = 300 N o Constant Velocity Dynamic equilibrium applies in states of constant, nonzero velocity. Zero net force A freebody diagram shows the forces. Zero net force used here: FN + Fg + Fy = 0 Fx + Ffr = 0 o Force at Impact A falling ball exerts a force when it strikes. The surface exerts a force to make the ball rebound. There are forces both ways at contact. Third Law: Law of Reaction For every action there is an equal and opposite reaction. Forces between two objects act in pairs. F12 = F21 (Newton’s third law) o Equal and Opposite Newton’s law of reaction also applies to the force of gravity. The Earth pulls the Moon The Moon pulls the Earth Newton used this to describe a Law of Gravity. o Universal Gravity Newton realized that all objects obey that Law. Other planets Apples People The gravitational force is universal. The gravitational constant is G = 6.67 x 1011 Nm2/kg2. What is the gravitational force between two students sitting in adjacent seats? Assume the students have a mass of 70 kg each. Assume that they are separated by 1m. F = (6.67 x 1011 Nm2/kg2) x (70 kg)2 / (1 m)2 F = 3.3 x 107 N. o Surface Gravity The force of gravity on a mass is its weight. The force of gravity compared to the mass is the field strength. Consider the force on an unknown mass m. Use Earth’s mass M and Earth’s radius R Calculate the field strength. g = F/m = GM/R2 M = 5.97 x 1024 kg R = 6.37 x 106 m g = 9.81 N/kg = 9.81 m/s2 This field strength g only applies at Earth’s surface. Approximately, g is equal to 9.8 N/kg or 9.8 m/s2. o Variations Gravity varies over the surface of the Earth. The height of the surface varies – so the radius does, too The material under the surface is not uniform The earth isn’t exactly round The tides affect the earth as well as the oceans One unit of gravitational acceleration used on the Earth is the Galileo. 1 Gal = 1 cm/s2 = 0.01 m/s2 = 0.01 N/kg. g = 981 Gal o Effect of Height High areas have a greater distance from the center of the Earth. An increase of 1 km should decrease g by 300 mGal. Type of rock affects g. o Latitude Changes As the Earth spins the equator slightly bulges. The radius is about 22 km bigger compared to the pole. Expect a few Gal difference from equator to pole. Equator: 9.780 m/s2 North Pole: 9.832 m/s2 o Earth’s Rotation The earth is made of layers of different types of rock. These rocks can move due to daily tides. Much less than ocean tides Period is 12 hours like the ocean The force of gravity from the Moon is countering some of the force of the Earth. o Gravity at Rest The force of gravity acts on all objects all the time. If an object is at rest, the law of inertia says that the net force is zero. There must be a force opposite to gravity that cancels it out. o Normal Force The force that opposes gravity for objects on the ground is called the normal force. It is perpendicular (normal) to the plane of the ground. The force is a result of the law of reaction. o Normal Force and Weight The normal force pushing up against gravity can be measured. We measure weight with a scale that measures normal force. Weight is a force, not a mass. Pounds measure weight, so force (not mass) can be measured in pounds. o Forces on a Slope Draw the forces acting on the block. The force of gravity points down with magnitude Fg=mg. The normal force FN points away from the surface of the inclined plane. An applied force Fappolds the block in place. o Net Force The block stays in place on the plane. No net force along the surface No net force away from the surface Find the components of gravity in those directions. Find the unknown forces o Mechanical Advantage The inclined plane has been used as a machine for centuries. Pushing a block up an incline is easier than lifting. The ratio of the force to lift directly compared to the force using the incline is the mechanical advantage. o Tension Force A taut rope has a force exerted on it. If the rope is lightweight and flexible the force is uniform over the entire length. This force is called tension and points along the rope. Tension acts on both ends. o Tension or Normal Force Tension and normal forces are different. A pull on an object tension A push from a surface normal force Either one or both may be present. o Equilibrium in One Dimension Two weights are hung supported by strings. On the lower block the two forces balance: FT2 = m2g On the upper block there are three: FT1 = m1g + FT2 FT1 = (m1 + m2)g The upper string has more tension than the lower string. o Pulley A pulley uses tension to transfer a force to another direction. Mechanical Advantage With more than one pulley the force needed to lift an object can be reduced. The pulley is a simple machine. The mechanical advantage to the left is 2. o Holding in Place Objects on an incline will often stay put. There must be a force that holds the object in place. Static friction is from the contact of resting objects. Force holds up to a certain point Force is based on the type of contact (rough, smooth) Maximum force is proportional to the pressing force of the object (normal force) o Inequality The approximate formula for static friction is: ms is the coefficient of static friction This is an inequality. The force of static friction is generally less than the coefficient times the normal force o Coefficient of Friction Use rope and measure the force when movement begins. Measure weight Measure force at slipping point. o Slippery Slope If Ffr < msFN = msmg cosq, then the block will hold. At equality the block just begins to move. o Sliding Sliding objects also have a frictional force exerted on them. This frictional force is kinetic friction. An approximate formula: mk is the coefficient of kinetic friction o Static vs. Kinetic Static and kinetic friction are similar. Force in opposite direction to motion Proportional to normal force Coefficient of friction depends on materials Static and kinetic friction are different. Static friction is an inequality up to a maximum Coefficient of friction is typically greater for static friction at the same surface Wood on wood (0.4 vs. 0.2) o Downhill Skiing A TV station has invited you to be the science commentator for an upcoming ski race. You observe that a skier needs at least 3° of slope to move forward at a constant velocity. o Force Diagram At constant velocity the forces must all balance. Friction doesn’t act in the direction of the normal force. The normal force cancels the component of gravity. o Minimum Sliding When the skier is sliding slowly we can neglect air resistance. Kinetic friction balances the component of gravity pulling forward. Normal Force and Friction Friction depends on both the normal force and on the coefficient of friction. To reduce friction requires reducing one of those factors. Reduce normal force by lightening the load Reduce normal force by adding additional upward force Add a lubricant to reduce the coefficient of friction Unit 3 o Defining Position Position has three properties: Origin, magnitude, direction o Position Graph Position can be displayed on a graph. The origin for position is the origin on the graph. Axes are position coordinates. The position is a vector. A set of position points connected on a graph is a trajectory. o Scalar Multiplication A vector can be multiplied by a scalar. Change feet to meters. Walk twice as far in the same direction. Scalar multiplication multiplies each component by the same factor. The result is a new vector, always parallel to the original vector. o Reference Point Displacement is different from position Position is measured relative to an origin common to all points. Displacement is measured relative to the object’s initial position. The path (trajectory) doesn’t matter for displacement. o Displacement Vector The position vector is often designated by . A change in a quantity is designated by Δ (delta). Always take the final value and subtract the initial value. o Two Displacements A hiker starts at a point 2.0 km east of camp, then walks to a point 3.0 km northeast of camp. What is the displacement of the hiker? Each individual displacement is a vector that can be represented by an arrow. o Vector Subtraction To subtract two vectors, place both at the same origin. Start at the tip of the first and go to the tip of the second. o Component Subtraction Multiplying a vector by 1 will create an antiparallel vector of the same magnitude. Vector subtraction is equivalent to scalar multiplication and addition. o Displacement Components Find the components of each vector, and subtract. Ax = 2.0 km Ay = 0.0 km Bx = (3.0 km)cos45 = 2.1 km By = (3.0 km)sin45 = 2.1 km Dx = Bx – Ax = 0.1 km Dy = By – Ay = 2.1 km o Speed Speed measures the rate of change of position along a path. The direction doesn’t matter for speed, but path does. o Velocity Velocity measures the rate of change of position, and specifies the direction. The rate of change of position is similar for speed and velocity. o Velocity Magnitude Displacement divided by time gives the average velocity. The displacement vector divided by time gives the average velocity vector. A person walks from 2 km north of the gym to 3 km west of the gym in 1.5 h. What is the magnitude of average velocity? The magnitude of displacement is The average velocity is the magnitude of displacement divided by the time. o Graphing Motion Kinematics is the study of motion. Motion requires a change in position. We graph position as a function of time. Which point is farthest? Which point is fastest? Which point is slowest? o Motion in One Dimension In one dimension objects only move on a straight line. Objects can go forward and backward. Objects can speed up and slow down. o Average Velocity The ratio v = Dx / Dt gives the average velocity during the time interval Dt. In the graph, x = x2 – x1 t = t2 – t1 This is the slope of the line. o Short Times If the time scale changes the velocity may also change. The average velocity from P1 to P2 (green) is greater than the average velocity from P1 to PA (orange). o Limits Mathematical limit describes the result of taking a value to an extreme. This is used when the result at the extreme point would give a mathematical expression that cannot be calculated. o Instantaneous Velocity The limit of the velocity for short times is the instantaneous velocity. The instantaneous velocity appears as the slope of the graph at a point. o Motion in Two Dimensions Time is a scalar, not a vector. o Tangent The average velocity becomes the instantaneous velocity for short time intervals. The same is true for vectors. The instantaneous velocity vector direction is tangent to the curve. o Changing Velocity In complicated motion the velocity is not constant. We can express a time rate of change for velocity just as for position, Dv = v2 v1. The average acceleration is the time rate of change of velocity: a = Dv / Dt. o Average Acceleration Example problem A jet plane has a takeoff speed of 250 km/h. If the plane starts from rest, and lifts off in 1.2 min what is the average acceleration? a = Dv / Dt = [(250 km/h) / (1.2 min)] * (60 min/h) a = 1.25 x 104 km/h2 Why is this so large? Is it reasonable? Does the jet accelerate for an hour? o Instantaneous Acceleration Instantaneous velocity is defined by the slope. Instantaneous acceleration is also defined by the slope. o Velocity to Position Area under a velocity curve equals the change in position. Acceleration to Velocity Area under an acceleration curve equals the change in velocity. Negative area is a decrease in value. o Velocity in Two Dimensions Position graph with velocity vectors. Velocity graph using an origin with zero speed. o Acceleration in Two Dimensions The acceleration shows the change in velocity. Acceleration, velocity and position may not line up. o Vector Equations Like velocity, acceleration equations can be written by components. o Mass and Force Mass is substance. Mass is measured in kilograms. 1 kg = 1000 g 1 atomic mass unit (about the mass of one hydrogen atom) = 1.66 x 1027 kg Force is measured in N. 1 N = 1 kg m/s2 Dimension M(L/T2) Pounds measure weight (a force) not mass. o Second Law: Law of Acceleration The change in motion is proportional to the net force and the change is made in the same direction as the net force. Net force gives rise to acceleration. Force = mass x acceleration (Newton’s second law). o Force and Acceleration With force there is acceleration. The amount of acceleration depends on the mass. Mass is a scalar. Mass times a vector gives another vector. The direction of the force and acceleration are the same. o Vector Force A 1000 kg satellite in space is moving at 5.0 km/s when a rocket fires with a thrust of 5.0 x 103 N at 60° to the direction of motion. The rocket fires for 1 minute. Where does it move after firing? Identify the quantities in the problem: Mass, m = 1000 kg Initial velocity, v0 = 5 x 103 m/s Force, F = 5 x 103 N at 60° Time, t = 1 min = 60 s Force and velocity are vectors Pick x in the direction of initial motion: vx = v0 . Fx = F cosq Fy = F sinq o Change in Velocity A 1000 kg satellite in space is moving at 5.0 km/s when a rocket fires with a thrust of 5.0 x 103 N at 60° to the direction of motion. The rocket fires for 1 minute. Where does it move after firing? The force gives acceleration. ax = Fx /m = (F/m) cosq ay = Fy /m = (F/m) sinq The change in velocity is due to this acceleration. vx = v0 + ax Dt = v0 + (F t / m) cosq vy = ay Dt = (F t / m) sinq The final velocity is vx = 5200 m/s = 5.2 km/s vy = 260 m/s = 0.26 km/s. o Unbalanced Forces Newton’s Second Law uses the net force. One or more forces act on an object Forces are vectors that can be added Add all vector forces acting on a object. If the sum is zero – equilibrium. If the sum is not zero – acceleration. o Force Vectors Force is a vector. A block sliding on an inclined plane has forces acting on it. We know there is a force of gravity and normal force. o Force Diagram Draw the forces acting on the block. The force of gravity points down with magnitude Fg=mg. The normal force points away from the surface of the inclined plane. Draw components of the forces so the vectors can be added. The coordinates should point along the surface. The normal force is unknown The motion will be along the surface The components of Fg are compared to the surface. o Net Force The block stays along the plane. No motion or acceleration away from the surface No force in that component Solve for the normal force: Solve for the net force: o Motion from Force The net force causes the block to accelerate. The amount of acceleration is force divided by mass. This is a constant acceleration moving the block. o Inclined Plane Pushing a block up an incline is easier than lifting. To get constant velocity a force equal to the downward acceleration must be applied. o Point of View A plane flies at a speed of 200. km/h relative to still air. There is an 80. km/h wind from the southwest (heading 45° north of east). What direction should the plane head to go due north? What is the speed of the plane relative to the ground? o Different Motion We need an end velocity in the direction of due north. Assign E to x and N to y. The wind velocity and the plane velocity must add to get the result. o Velocity by Components The velocity of the wind can be described in the ground’s coordinates. wx = (80. km/h) cos 45° = 57 km/h wy = (80. km/h) sin 45° = 57 km/h The velocity of the plane is also described compared to the ground as if in still air. px = p cos q py = p sin q o Velocity Vector Sum The plane’s net motion compared to the ground is the sum of the wind velocity and plane velocity. vx = wx + px = wx + p cos q vy = wy + py = wy + p sin q The plane should only go north, so vx = 0. wx = p cos q 57 km/h = (200. km/h) cos q cos q = 0.285, or q = 106.6° ? 110° compared to +x axis o View from the Ground The relative velocity in air was given. The angle was found. Finally the ground speed is found. Simplified since the plane is headed north compared to the ground. o Reference Frame Displacement is different from position The displacement is measured relative to the object’s current position. Velocity can be measured relative to the object’s current velocity. This is the relative velocity. Example: walking on a moving train. Some measurements may be taken in different reference frames. From the example: ground, plane, wind Unit 4 o Graphs to Functions A simple graph of constant velocity corresponds to a position graph that is a straight line. The functional form of the position is This is a straight line and only applies to straight lines. o Constant Acceleration Constant velocity gives a straight line position graph. Constant acceleration gives a straight line velocity graph. The functional form of the velocity is o Top Fuel A top fuel dragster can reach 150 m/s (330 mph) in 3.7 s. What is the acceleration? In the equation v0 = 0 The acceleration is v/t. a = 41. m/s2 o Acceleration and Position For constant acceleration the average acceleration equals the instantaneous acceleration. Since the average of a line of constant slope is the midpoint: o Acceleration Relationships Algebra can be used to eliminate time from the equation. This gives a relation between acceleration, velocity and position. For an initial or final velocity of zero. This becomes x = v2 / 2a v2 = 2 a x o Drag Strip A top fuel dragster accelerates at 40. m/s2 for 3.7 s. How far does the car go on the track? Starts at rest and at 0. x = 270 m o Accelerating a Mass A loaded 747 jet has a mass of 4.1 x 105 kg and four engines. It takes a 1700 m runway at constant thrust (force) to reach a takeoff speed of 81 m/s (290 km/h). What is the force per engine? The distance and final velocity are used to get the acceleration. The acceleration and mass give the force. o Falling on the Earth Items that start by going up, stop and then fall again, reversing direction. More height more speed The velocity changes, there must be an acceleration. o Surface Gravity At the surface of the Earth, gravity creates a nearly constant acceleration. This gravitational acceleration is called g. g = 9.8 m/s2 and is directed downward. Note that g is an acceleration. Don’t be fooled by phrases like gforce. g can be used as a unit of acceleration. o Pulling g’s A top fuel dragster can reach 150 m/s (330 mph) in 3.7 s. What is the acceleration in g’s? The acceleration is v/t. a = 40. m/s2 Convert units. a = 4.0 g Up and Down The constant acceleration equations apply for a = g. The position is now y, in the vertical direction. o Initial Velocity A person throws a ball up in the air releasing the ball at 1.5 m. If it lands on the ground after 2.0 s, find the initial velocity. Simplify the time equation, with y0 = 1.5 m y = 0 m t = 2.0 s. o Exceptions to Free Fall On Earth’s surface downward acceleration is constant. Why do we observe that different objects accelerate differently? Friction reduces the total acceleration. We think about velocity instead of acceleration. o Pulley Acceleration The normal force on m1 equals the force of gravity. The force of gravity is the only external force on m2. Both masses must accelerate together. o Atwood’s Machine In an Atwood machine both masses are pulled by gravity, but the force is unequal. o Acceleration of Gravity Objects that fall to the Earth all experience an acceleration. The acceleration due to gravity is g = 9.8 m/s2. This acceleration must be due to a force. o Force of Gravity The acceleration of a falling mass m is g. The force on the mass is found from F = ma (action). This gravitational force is F = mg. o Normal Weight We measure weight with a scale that measures normal force. W = mg Weight is related to mass by the gravitational field g. o False Weight A vertical acceleration can change the weight. The normal force on the floor is our sense of weight. Downward acceleration reduces weight Upward acceleration increases weight Mass is unchanged. o Newton’s law of acceleration F = ma net force, F = mg + FN. Solve for the normal force mg + FN = ma FN = ma + mg FN = m (a + g) Apparent mass based on g mapp = FN / g o Accelerated Weight An elevator is accelerating downward at 2.0 m/s2. The person has a mass of 70 kg. What mass is on the scale? Add all the forces, but the net force is – ma = FN – mg. Solve for FN = m (g – a) Convert to mass mapp = FN /g The scale shows 56 kg. o Weightlessness If the elevator accelerated downward at g, the normal force would become 0. FN = m (g – a) = m (g – g) = 0 The person would feel weightless. An object in free fall is weightless, but not massless. Microgravity NASA studies weightlessness here on the Earth. Use free fall o Horizontal and Vertical Motion Position, velocity and acceleration are vectors. These vectors can be separated into components. Choose x for horizontal Choose y for vertical The equations for constant acceleration can be written separately in each coordinate. o Vector Gravity The gravitational acceleration is a vector. The magnitude is 9.8 m/s2. The direction is down. If there is only constant gravitational acceleration in the y direction, the equations simplify for x. Shoot the Monkey A smart monkey and smart hunter – who wins? o Vertical Pull All falling objects are subject to the same gravitational acceleration. The horizontal component doesn’t affect the downward acceleration. o Moving Target The initial horizontal velocity doesn’t matter. All objects fall at the same vertical rate (neglecting friction). At the collision the target fell a distance y = d. The time it fell comes from the acceleration. o Horizontal Speed The point of collision measures the distance dropped since the ball was fired. The time determines the horizontal velocity of the ball. The angle of the launcher gives the total velocity. o Projectile Path All projectiles follow the same shape path. This path is a parabola. Acceleration is constant. Horizontal velocity is also constant. Velocity and position change. o Parameterized Curve The two equations of motion can be combined to give an equation for the trajectory. We’ve set the origin to x0 = 0, y0 = 0. o Height The maximum height occurs when vy = 0. This is identical to 1dimensional motion. From the height can the initial velocity be estimated? o Fireworks A fireworks shell is shot into the air with an initial speed of 70.0 m/s at an angle of 75.0º above the horizontal. What is it’s maximum height? vy0 = 67.6 m/s y = 233 m o Range The range of a projectile is the horizontal distance. The simplest range to find is for a projectile returning to its initial height. This is linked to the angle of the initial velocity. o Dud A fireworks shell is shot into the air and fails to ignite. Initial speed of 70.0 m/s Angle of 75.0º above the horizontal How far away does the shell land? x = 250. m o Time The time in flight can be determined by height alone. The “hang time” is just this time in flight. A 1 m jump will be in the air for about 0.9 s. The range doesn’t matter. o Ignition A fireworks shell needs a fuse that will ignite at the peak height. Initial speed of 70.0 m/s Angle of 75.0º above the horizontal Height 233 m How much time should the fuse have? Up only t = 6.90 s o Different Heights Not all motion starts and stops at the same height. The difference in height can be found by the full set of equations. Let y be some nonzero value. Unit 5 o Position on a Circle Motion in a circle is common. The most important measure is the radius (r). The position of a point on the circle is described by a radial vector . Origin is at the center. Magnitude is equal everywhere. o Measuring a Circle We use degrees to measure position around the circle. There are 2p radians in the circle. This matches 360° The distance around a circle is s = r q, where q is in radians. o Period and Frequency Movement around a circle takes time. The period (T) is the time it takes to complete one revolution around the circle. The frequency (f) is the number of cycles around completed in a time. Cycles per second (cps or Hz) Revolutions per minute (rpm) Frequency is the inverse of period (f = 1/T). o Cycles or Radians Frequency is measured in cycles per second. There is one cycle per period. Frequency is the inverse of the period, f =1/T. Angular velocity is measured in radians per second. There are 2p radians per period. Angular velocity, w = 2p/T. Angular velocity, w = 2pf. o Angular Velocity Displacement is related to the angle. Displacement on the curve (s) Angle around the circle (q) Velocity has an angular equivalent. Linear velocity (v) Angular velocity (w) Units (rad/s or 1/s = s1) o Speed on a Circle The circumference of a circle is 2pr. The period is T. The speed is related to the distance and the period or frequency. v = 2pr/T v = 2prf v = rw o Velocity on a Circle Velocity is a vector change in position compared to time. As the time gets shorter, the velocity gets closer to the tangent. o Direction of Motion In the limit of very small angular changes the velocity vector points along a tangent of the circle. This is perpendicular to the position. For constant w, the magnitude stays the same, but the direction always changes. o No Slipping A wheel can slide, but true rolling occurs without slipping. As it moves through one rotation it moves forward 2pR. o Point on the Edge A point on the edge moves with a speed compared to the center, v = wr. Rolling motion applies the same formula to the center of mass velocity, v = wR. The total velocity of points varies by position. o Acceleration in a Circle Acceleration is a vector change in velocity compared to time. For small angle changes the acceleration vector points directly inward. This is called centripetal acceleration. o Centripetal Acceleration Uniform circular motion takes place with a constant speed but changing velocity direction. The acceleration always is directed toward the center of the circle and has a constant magnitude. o Buzz Saw A circular saw is designed with teeth that will move at 40. m/s. The bonds that hold the cutting tips can withstand a maximum acceleration of 2.0 x 104 m/s2. Find the maximum diameter of the blade. Start with a = v2/r. r = v2/a. Substitute values: r = (40. m/s)2/(2.0 x 104 m/s2) r = 0.080 m. Find the diameter: d = 0.16 m = 16 cm. o Law of Acceleration in Circles Motion in a circle has a centripetal acceleration. There must be a centripetal force. Vector points to the center The centrifugal force that we describe is just inertia. It points in the opposite direction – to the outside It isn’t a real force o Conical Pendulum A 200. g mass hung is from a 50. cm string as a conical pendulum. The period of the pendulum in a perfect circle is 1.4 s. What is the angle of the pendulum? What is the tension on the string? o Radial Net Force The mass has a downward gravitational force, mg. There is tension in the string. The vertical component must cancel gravity FTy = mg FT = mg / cos q FTr = mg sin q / cos q = mg tan q This is the net radial force – the centripetal force. o Acceleration to Velocity The acceleration and velocity on a circular path are related. o Period of Revolution The pendulum period is related to the speed and radius. o Radial Tension The tension on the string can be found using the angle and mass. FT = mg / cos q = 2.0 N If the tension is too high the string will break! o Force on a Curve A vehicle on a curved track has a centripetal acceleration associated with the changing direction. The curve doesn’t have to be a complete circle. There is still a radius (r) associated with the curve The force is still Fc = mv2/r directed inward o Friction on a Wheel A rolling wheel does not slip. It exhibits static friction. As a car accelerates the tire pushes at the point of contact. The ground pushes back, accelerating the car. o Curves and Friction On a turn the force of static friction provides the centripetal acceleration. In the force diagram there is no other force acting in the centripetal direction. o Skidding The limit of steering in a curve occurs when the centripetal acceleration equals the maximum static friction. A curve on a dry road (ms = 1.0) is safe at a speed of 90 km/h. What is the safe speed on the same curve with ice (ms = 0.2)? 90 km/h = 25 m/s rdry = v2/ ms g = 64 m v2icy = ms g r = 120 m2/s2 vicy = 11 m/s = 40 km/h o Banking Curves intended for higher speeds are banked. Without friction a curve banked at an angle q can supply a centripetal force Fc = mg tan q. The car can turn without any friction. o Hill Top A car going over a hill follows a vertical curve. There are only two forces acting on the moving car. Gravity Normal force If the normal force is zero the car has lost contact with the ground. o Speed Bump A neighborhood speed bump has a radius of 1 ft. What is the maximum speed (mph) a car can travel without flying off the bump? The normal force is zero is the car goes airborne. 1 ft = 12*2.54 cm = 30 cm 30 cm = 0.3 m The maximum speed is 1.7 m/s. 1.7 m/s * 3600 s/h / 1000 m/km = 6.2 km/h 6.2 km/h / 1.6 km/mi = 3.9 mi/h This is 4 mph. o LooptheLoop A looptheloop is a popular rollercoaster feature. There are only two forces acting on the moving car. Gravity Normal force There is a centripetal acceleration from the curve. o Staying on Track If the normal force becomes zero, the coaster will leave the track in a parabolic trajectory. Projectile motion At any point there must be enough velocity to maintain pressure of the car on the track. o Force at the Top The forces of gravity and the normal force are both directed down. Together these must match the centripetal force. The minimum occurs with almost no normal force. o Centrifuge A centrifuge spins to create an artificial gravity. Liquid separation Pilot training Amusement rides o Fake Gravity The centripetal force is like an elevator accelerating upward. Fnet = mar = FN The net force must be due to a normal force. Experience as weight If rw2 = g then it matches earth’s gravity. o Gravitron The Gravitron is an amusement ride that uses centripetal force. Rotates at 24 rpm Diameter is 14 m o Centripetal Correction The spin of the earth creates a centripetal acceleration toward the axis of the earth. The effect is like accelerating down in an elevator. Items weigh less at the equator. o Orbital Force Newton’s law of acceleration applies to orbits. Centripetal acceleration times mass is the force of gravity The situation is the same for the satellite or any object in the satellite. o Free Fall The net force of an object in circular orbit matches the centripetal acceleration. This is the same for a freely falling object. Velocity does not change the force or acceleration. o Weightlessness Objects in free fall exert no normal force. Fnet = ma = mg + FN If a = g, FN = 0 The same is true in orbit. Fnet = mar = Fgrav + FN If ar = Fgrav/m , FN = 0 Objects in orbit are weightless. o Tidal Gravity The Moon’s gravity is strongest at the closest point. Near face ocean pulled most Far face ocean pulled least Sides pulled slightly in o Ocean Tides From Earth a person doesn’t sense the Moon’s pull. Near and far faces of the ocean bulge away Sides are pulled slightly in Tides occur twice a day as Earth rotates. o Pendulum Swing On a turntable a pendulum doesn’t appear to swing straight. Table turns the observer Fictitious force The deflection is called the Coriolis force. o Cyclone In the absence of rotation air would move from high to low pressure. Straight line wind The Coriolis force causes wind to turn. Friction causes equilibrium Circular wind pattern around low o Short Period An object in space would go in a straight line without another force. Gravity supplies a force to hold objects in circular orbits. In low orbit the period is related to the gravitational acceleration. o Geosynchronous Orbit In higher orbits, the gravitational force is significantly less than on the surface. Use the force of universal gravitation. Fgrav = G M m / r2 The height for a satellite with a 24 hr period can be found. o Testing Models Geocentric (or Ptolemaic) means the Earth is at the center and motionless. Heliocentric (or Copernican) means the Sun is at the center and motionless. Scholars wanted to differentiate models by comparing the predictions with precise observations. This originated the modern scientific method. o Kepler’s Work Tycho Brahe led a team which collected data on the position of the planets (1580 1600 with no telescopes). Mathematician Johannes Kepler was hired by Brahe to analyze the data. He took 20 years of data on position and relative distance. No calculus, no graph paper, no log tables. Both Ptolemy and Copernicus were wrong. He determined 3 laws of planetary motion (16001630). o Kepler’s First Law The orbit of a planet is an ellipse with the sun at one focus. o Orbital Speed The centripetal force is due to gravity. GMm/r2 = mv2/r v2 = GM/r Larger radius orbit means slower speed. Within an ellipse larger distance also gives slower speed. o Kepler’s Second Law The line joining a planet and the sun sweeps equal areas in equal time. o Orbital Period An ellipse is described by two axes. Long – semimajor (a) Short – semiminor (b) The area is pab (becomes pr2 for a circle). The speed is related to the period in a circular orbit. v2 = GM/r (2pr/T)2 = GM/r T2 = 4p2r3/GM Larger radius orbit means longer period. Within an ellipse, a larger semimajor axis also gives a longer period. o Kepler’s Third Law The square of a planet’s period is proportional to the cube of the length of the orbit’s semimajor axis. T2/a3 = constant The constant is the same for all objects orbiting the Sun Unit 6 o Force over a Distance In physics, work is the force acting on object over a distance. Forces acting on objects at rest do no work. o Creating Motion Work is force acting over a distance. Changing motion requires a force: F = ma. Force is applied over a finite distance. Forces that change motion may do work. o Work in a Direction The force must act in the direction of the displacement to be work. Beginning and End Only the force acting in the direction of displacement counts towards work. Displacement considers the beginning and end points. Forces acting perpendicular to an object’s motion do no work. Linear motion with perpendicu
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