Physics 1 First Semester Review Cheat Sheet, Cheat Sheet of Physics

Download Physics 1 First Semester Review Cheat Sheet and more Cheat Sheet Physics in PDF only on Docsity! A.P. Physics 1 First Semester Review Sheet Fall, Dr. Wicks Chapter 1: Introduction to Physics • Review types of zeros and the rules for significant digits • Review mass vs. weight, precision vs. accuracy, and dimensional analysis problem solving. Chapter 2: One-Dimensional Kinematics A. Velocity • Equations for average velocity: f iave f i x xxv t t t -D = = D - and 1 ( ) 2ave f i v v v= + • In a position-versus-time graph for constant velocity, the slope of the line gives the average velocity. See Table 1. • Instantaneous velocity can be determined from the slope of a line tangent to the curve at a particular point on a position-versus-time graph. • Use vave = ΔxTotal ΔtTotal to calculate the average velocity for an entire journey if given information about the various legs of the journey. B. Acceleration • Equation for average acceleration: f iave f i v vva t t t -D = = D - • In a velocity-versus-time graph for constant acceleration, the slope of the line gives acceleration and the area under the line gives displacement. See Table 1. • Acceleration due to gravity = g = 9.81 m/s2. (Recall a = - g = -9.81 m/s2) Table 1: Graphing Changes in Position, Velocity, and Acceleration Constant Position Constant Velocity Constant Acceleration Ball Thrown Upward Position Versus Time: Velocity Versus Time: Accelera -tion Versus Time: t x t x Slope = vave t x t x t v t v t v Slope = aave t v Slope = -9.81 m/s2 t a t a t a t a a = -9.81m/s2 A.P. Physics 1 First Semester Review Sheet, Page 2 Table 2: Comparing the Kinematic Equations Kinematic Equations Missing Variable o avex x v t= + a ov v at= + xD 21 2o o x x v t at= + + finalv 2 2 2ov v a x= + D tD Chapter 3: Vectors in Physics A. Vectors • Vectors have both magnitude and direction whereas scalars have magnitude but no direction. • Examples of vectors are position, displacement, velocity, linear acceleration, tangential acceleration, centripetal acceleration, applied force, weight, normal force, frictional force, tension, spring force, momentum, gravitational force, and electrostatic force. • Vectors can be moved parallel to themselves in a diagram. • Vectors can be added in any order. See Table 3 for vector addition. • For vector r  at angle q to the x-axis, the x- and y-components for r  can be calculated from Δx = r cosθ and Δy = r sinθ . • The magnitude of vector r  is r = Δx 2 + Δy2 and the direction angle for r  relative to the nearest x-axis is θ = tan−1 Δy Δx ⎛ ⎝⎜ ⎞ ⎠⎟ . • To subtract a vector, add its opposite. • Multiplying or dividing vectors by scalars results in vectors. • In addition to adding vectors mathematically as shown in the table, vectors can be added graphically. Vectors can be drawn to scale and moved parallel to their original positions in a diagram so that they are all positioned head-to-tail. The length and direction angle for the resultant can be measured with a ruler and protractor, respectively. B. Relative Motion • Relative motion problems are solved by a special type of vector addition. • For example, the velocity of object 1 relative to object 3 is given by v  13 = v  12 + v  23 where object 2 can be anything. • Subscripts on a velocity can be reversed by changing the vector’s direction: v  12 = −v  21 A.P. Physics 1 First Semester Review Sheet, Page 5 • The range of a projectile launched at initial velocity ov and angle q is 2 sin 2ovR g q æ ö = ç ÷ è ø • The maximum height of a projectile above its launch site is 2 2 max sin 2 ovy g q = Chapter 5: Newton’s Laws of Motion Table 5: Newton’s Laws of Motion Modern Statement for Law Translation Newton’s First Law: (Law of Inertia) Recall that mass is a measure of inertia. If the net force on an object is zero, its velocity is constant. An object at rest will remain at rest. An object in motion will remain in motion at constant velocity unless acted upon by an external force. Newton’s Second Law: An object of mass m has an acceleration a  given by the net force F  ∑ divided by m . That is a  = F  ∑ m netF ma= Newton’s Third Law: Recall action-reaction pairs For every force that acts on an object, there is a reaction force acting on a different object that is equal in magnitude and opposite in direction. For every action, there is an equal but opposite reaction. A. Survey of Forces • A force is a push or a pull. The unit of force is the Newton (N); 1 N = 1 kg-m/s2 • See Newton’s laws of motion in Table 5. Common forces on a moving object include an applied force, a frictional force, a weight, and a normal force. • Contact forces are action-reaction pairs of forces produced by physical contact of two objects. Review calculations regarding contact forces between two or more boxes. • Field forces like gravitational forces, electrostatic forces, and magnetic forces do not require direct contact. They are studied in later chapters. • Forces on objects are represented in free-body diagrams. They are drawn with the tails of the vectors originating at an object’s center of mass. • Weight, W  , is the gravitational force exerted by Earth on an object whereas mass, m , is a measure of the quantity of matter in an object (W mg= ). Mass does not depend on gravity. • Apparent weight, W  a , is the force felt from contact with the floor or a scale in an accelerating system. For example, the sensation of feeling heavier or lighter in an accelerating elevator. • The normal force, N  , is perpendicular to the contact surface along which an object moves or is capable of moving. Thus, for an object on a level surface, N  and W  are equal in size but opposite in direction. However, for an object on a ramp, this statement is not true because N  is perpendicular to the surface of the ramp. • Tension, T  , is the force transmitted through a string. The tension is the same throughout the length of an ideal string. A.P. Physics 1 First Semester Review Sheet, Page 6 • The force of an ideal spring stretched or compressed by an amount x is given by Hooke’s Law, F  x = −kx . Note that if we are only interested in magnitude, we use F kx= where k is the spring or force constant. Hooke’s Law is also used for rubber bands, bungee cords, etc. Chapter 6: Applications of Newton’s Laws A. Friction • Coefficient of static friction = ,maxSS F N µ = where F  S ,max is the max. force due to static friction. • Coefficient of kinetic friction = KK F N µ = where F  K is the force due to kinetic friction. • A common lab experiment involves finding the angle at which an object just begins to slide down a ramp. In this case, a simple expression can be derived to determine the coefficient of static friction: tanSµ q= . Note that this expression is independent of the mass of the object. B. Newton’s Second Law Problems (Includes Ramp Problems) 1. Draw a free-body diagram to represent the problem. 2. If the problem involves a ramp, rotate the x- and y-axes so that the x-axis corresponds to the surface of the ramp. 3. Construct a vector table including all of the forces in the free-body diagram. For the vector table’s column headings, use vector, x-direction, and y-direction. 4. Determine the column total in each direction: a. If the object moves in that direction, the total is ma . b. If the object does not move in that direction, the total is zero. c. Since this is a Newton’s Second Law problem, no other choices besides zero and ma are possible. 5. Write the math equations for the sum of the forces in the x- and y-directions, and solve the problem. It is often helpful to begin with the y-direction since useful expressions are derived that are sometimes helpful later in the problem. Recall that the math equations regarding friction and weight are often substituted into the math equations to help solve the problem. C. Equilibrium • An object is in translational equilibrium if the net force acting on it is zero, F  = 0∑ . • Equivalently, an object is in equilibrium if it has zero acceleration. • If a vector table is needed for an object in equilibrium, then F  x = 0∑ and F  y = 0∑ . • Typical problems involve force calculations for objects pressed against walls and tension calculations for pictures on walls, laundry on a clothesline, hanging baskets, pulley systems, traction systems, connected objects, etc. D. Connected Objects • Connected objects are linked physically, and thus, they are also linked mathematically. For example, objects connected by strings have the same magnitude of acceleration. • When a pulley is involved, the x-y coordinate axes are often rotated around the pulley so that the objects are connected along the x-axis. • A classic example of a connected object is an Atwood’s Machine, which consists of two masses connected by a string that passes over a single pulley. The acceleration for this system is given by 2 1 1 2 m ma g m m æ ö- = ç ÷+è ø . A.P. Physics 1 First Semester Review Sheet, Page 7 Chapter 7: Work and Kinetic Energy A. Work • A force exerted through a distance performs mechanical work. • When force and distance are parallel, W Fd= with Joules (J) or Nm as the unit of work. • When force and distance are at an angle, only the component of force in the direction of motion is used to compute the work: ( cos ) cosW F d Fdq q= = • Work is negative if the force opposes the motion (q >90o). Also, 1 J = 1 Nm = 1 kg-m2/s2. • If more than one force does the work, then 1 n Total i i W W = =å • The work-kinetic energy theorem states that 2 21 1 2 2Total f i f i W K K K mv mv= D = - = - • See Table 6 for more information about kinetic energy. • In thermodynamics, ( )A FW Fd Fd Ad P V A A æ ö æ ö= = = = Dç ÷ ç ÷ è ø è ø for work done on or by a gas. Table 6: Kinetic Energy Kinetic Energy Type Equation Comments Kinetic Energy as a Function of Motion: 21 2 K mv= Used to represent kinetic energy in most conservation of mechanical energy problems. Kinetic Energy as a Function of Temperature: 21 3 2 2aveave mv K kTæ ö = =ç ÷ è ø Kinetic theory relates the average kinetic energy of the molecules in a gas to the Kelvin temperature of the gas. B. Determining Work from a Plot of Force Versus Position • In a plot of force versus position, work is equal to the area between the force curve and the displacement on the x-axis. For example, work can be easily computed using W Fd= when rectangles are present in the diagram. • For the case of a spring force, the work to stretch or compress a distance x from equilibrium is 21 2 W kx= . On a plot of force versus position, work is the area of a triangle with base x (displacement) and height kx (magnitude of force using Hooke’s Law, F kx= ). C. Determining Work in a Block and Tackle Lab • The experimental work done against gravity, LoadW , is the same as the theoretical work done by the spring scale, ScaleW . • WOutput =WLoad = FdLoad =W  dLoad = mgdLoad where Loadd = distance the load is raised. • Input Scale ScaleW W Fd= = where F = force read from the spring scale and Scaled = distance the scaled moved from its original position. • Note that the force read from the scale is ½ of the weight when two strings are used for the pulley system, and the force read is ¼ of the weight when four strings are used. A.P. Physics 1 First Semester Review Sheet, Page 10 Chapter 10: Rotational Kinematics and Energy A. Rotational Motion • Angular position, , in radians is given by where is arc length and is radius. • Recall that θ(rad) = 2π radians 360o ⎛ ⎝⎜ ⎞ ⎠⎟ θ(deg) . • Counterclockwise (CCW) rotations are positive, and clockwise (CW) rotations are negative. • In rotational motion, there are two types of speeds (angular speed and tangential speed) and three types of accelerations (angular acceleration, tangential acceleration, and centripetal acceleration). See Table 9 for a comparison. • Since velocity is a vector, there are two ways that an acceleration can be produced: (1) changing the velocity’s magnitude and (2) changing the velocity’s direction. In centripetal acceleration, the velocity’s direction changes. • When a person drives a car in a circle at constant speed, the car has a centripetal acceleration due to its changing direction, but it has no tangential acceleration due to its constant speed. • The total acceleration of a rotating object is the vector sum of its tangential and centripetal accelerations. Table 9: Comparing Angular and Tangential Speed and Angular, Tangential, and Centripetal Acceleration Calculation Equations Units, Comments Angular Speed: • radians/s • Same value for horses A and B, side-by-side on a merry- go-round. Tangential Speed: • m/s • Different values for horses A and B, side-by-side on a merry-go-round. Angular Acceleration: • radians/s2 • Same value for horses A and B, side-by-side on a merry- go-round. Tangential Acceleration: • m/s2 • Different values for horses A and B, side-by-side on a merry-go-round. Centripetal Acceleration: • m/s2 • is perpendicular to with directed toward the center of the circle and tangent to it. qD s r q DD = sD r ave t qw D= D tv rw= ave t wa D= D ta ra= 2 2t c va r r w= = ca ta ca ta A.P. Physics 1 First Semester Review Sheet, Page 11 • Centripetal force, CF , is a force that maintains circular motion: 2 2t C C mvF ma mr r w= = = • The period, T , is the time required to complete one full rotation. If the angular speed is constant, then 2T p w = . • The equations for rotational kinematics are the same as the equations for linear kinematics. See Table 10 for a comparison. Table 10: Kinematic equations for Rotational Motion Linear Equations Angular Equations o avex x v t= + θ = θo +ω avet ov v at= + ω =ω o +αt 21 2o o x x v t at= + + θ = θo +ω ot + 1 2 αt2 2 2 2ov v a x= + D ω 2 =ω o 2 + 2α (Δθ ) • A comparison of linear and angular inertia, velocity, acceleration, Newton’s second law, work, kinetic energy, and momentum are presented in Table 11. • The moment of inertia, , is the rotational analog to mass in linear systems. It depends on the shape or mass distribution of the object. In particular, an object with a large moment of inertia is difficult to start rotating and difficult to stop rotating. See Table 10-1 on p.298 for moments of inertia for uniform, rigid objects of various shapes and total mass. • The greater the moment of inertia, the greater an object’s rotational kinetic energy. • An object of radius , rolling without slipping, translates with linear speed and rotates with angular speed . I r v v r w = A.P. Physics 1 First Semester Review Sheet, Page 12 Table 11: Comparing Equations for Linear Motion and Rotational Motion Measurement or Calculation Linear Equations Angular Equations Inertia: Mass, m where a constant Average Velocity: Average Acceleration: Newton’s Second Law: Work: Kinetic Energy: Momentum: Chapter 11: Rotational Dynamics and Static Equilibrium A. Torque • A tangential force, , applied at a distance, , from the axis of rotation produces a torque in Nm. (Since is perpendicular to , this is sometimes written as .) • A force applied at an angle to the radial direction produces the torque • Counterclockwise torques are positive, and clockwise torques are negative. • The rotational analog of force, , is torque, , where moment of inertia and angular acceleration. • The conditions for an object to be in static equilibrium are that the total force and the total torque acting on the object must be zero: , , . Related problems often involve bridges, scaffolds, signs, and rods held by wires. B. Solving Static Equilibrium Problems 1. Construct a diagram showing all of the forces. 2. Create an equation adding the forces together. • Remember to enter correct signs in your force equation. For example, upward forces are positive and downward forces are negative. • Since the object is not moving, set the force equation equal to zero. ( ) 2I kmr= k = ave xv t D = D ave t qw D= D ave va t D = D ave t wa D= D netF ma= I F rt a ^= = cosW Fd q= W tq= 21 2 K mv= 21 2 K Iw= p mv= L Iw= F r rFt = F r F rt ^= sinrFt q= F ma= It a= I = a = 0xF =å 0yF =å 0t =å F = 0∑