MCAT Physics Review/Summary
MCAT Physics Review/Summary
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Date Created: 01/13/15
Physics 7C Summary November 24 2013 Contents 1 Essentials 7 11 Multiplying Objects 7 111 Dot Product 7 112 Cross Product 8 12 Right Hand Rule 8 13 Determinants for Cross Products 8 14 E to f 9 15 Choosing Coordinate Axes 9 2 Kinematics 11 21 Average Velocity and Average Acceleration 11 22 Instantaneous Velocity and Instantaneous Acceration 11 23 Four Big Kinematic Equations for CONSTANT ACCELERATION 11 3 Forces Newton s Laws of Motion 13 31 Newton s Three Laws of Motion 13 32 Free Body Diagrams 13 321 Gravitational Forces 13 322 Normal Forces 13 323 Frictional Forces 13 324 Tension Forces 14 325 Spring Forces 14 33 Static Equilibrium and Dynamics 14 331 Static Equlibrium 14 332 Dynamics 14 4 Circular Motion 15 41 Centripetal Acceleration 15 5 Work Energy Energy Conservation 17 51 Types of Energy 17 511 Kinetic Energy 17 512 Potential Energy 17 513 Work 17 52 Work Energy Theorem 18 53 Conservation of Energy 18 4 CONTENTS 6 Momentum Impulse and Collisions 19 61 Momentum 19 62 Conservation of Momentum 19 63 Collisions 19 631 Elastic Collisions 19 632 Inelastic Collisions 20 64 Solving Collision Problems 20 7 Center of Mass 21 8 Rotational Kinematics Moment of Inertia 23 81 Rotational Analogs 23 82 Rotational Kinematics 23 821 Rotational Kinematics under CONSTANT ANGULAR ACCELERATION 24 822 Linear to Rotational Transformations 24 823 Total Acceleration from Moving around in a Circle 24 824 Period from Moving around in a Circle 24 83 Energy from Moving in Circular Motion 24 84 Moment of lneria 25 85 Parallel Axis Theorem 25 86 Rolling Without Slipping 25 9 Torque and Rotational Dynamics 27 91 Torque 27 92 Work in Rotational Motion 27 93 Rotation Linear Motion 27 931 Kinetic Energy of Rolling 27 10 Angular Momentum 29 101 Angular Momentum 29 102 Conservation of Angular Momentum 29 103 When there are External Forces 29 Introduction Hiya Just a review of the course D CONTENTS Chapter 1 Essentials 11 Multiplying Objects There are different ways to multiply things together depending on what we are multiplying We are so used to multiplying numbers together such as 5 X 7 that it has been ingrained in us that multiplication is just that But what we have been doing is multiplying numbers called scalars We can actually multiply different objects that are not scalars The only other objects you would have to deal with are vectors De nition 11 A vector is an object that is a scalar and has a direction Picture it as an arrow that has a head and a tail The head is in the direction the vector is pointing Think of it this way Let s consider what we get when we multiply scalars together Scalars are determined by just one property a value Because of this ONE property when we multiply we should get something with this ONE property back Now when we deal with vectors there are two properties a value and a direction So when we multiply we should expect to get something that can have up to TWO properties We can multiply and get something that is JUST a value scalar Or we can multiply and get something that is both a value and a direction vector There are only two you have to worry about They are called the dot product and the cross product 111 Dot Product The dot product gives back a SCALAR It multiplies two vectors and spits out a scalar De nition 12 Given two vectors if and g the dot product is de ned as II E ABc086 where 6 is the angle formed by putting the two vectors with their tails touching A lot of people ask what is the meaning of the dot product and it essentially tells you how parallel the two vectors are When the dot product is large then the two vectors are closer together in terms of angles and the angle between them is small When the dot product is small the two vectors are far apart in terms of angles and the angle between them is large 7 8 CHAPTER 1 ESSENTIALS 112 Cross Product The cross product gives back a VECTOR It multiplies two vectors and spits out a vector De nition 13 Given two vectors if and IS the cross product is de ned as if X IS AB8in6 where 6 is the angle formed by putting the two vectors with their tails touching The resulting vector is perpendicular to BOTH ff AND IS Imagine if and IS on a piece of blank paper on a table Their cross product points directly out of the page perpendicular to the paper and table The cross product tells you how perpendicular two vectors are When the cross product is small then the two vectors are closer together in angle and less perpendicular When the cross product is large then the two vectors are farther apart in angle and more perpendicular 12 Right Hand Rule The right hand rule is just a way to help you nd the direction from a cross product Let s say I was given the cross product if X E To use the right hand rule your ngers represent if and the palm of your hand represents 5 You rst point your ngers along Then make your palm face the direction of IS That s it depending on the context your thumb represents the direction of the cross product or for rotations your ngers curl in the direction you are looking for 13 Determinants for Cross Products Instead of using the above equation for the cross product you can use the following if they give you the components of the vectors instead Let s say they gave you if lt am ay 02 gt and I lt bx by b2 gt we can nd the cross product of these two as am ay a2 bx by b2 which can be solved like this Start at 5 and cross out that row and column like this 3339 5 A 33 33 a ay 02 am y 02 am ay 2 m b3 b2 bx y b2 bag by b What you are now left with are 3 smaller 2562 determinants To get the cross product you put these 3 determinants as so ay 02 by 2 am 02 bx b2 am ay bx by A y A 14 2 To 9 abe azby ame azbm amby aybw And that s your cross product Remember the cross product is a vector 14 Zto f Physics will always talk about individual particles rst and then move on to objects that are continuous The individual particles are like dots on a page that have no real dimension to them and continuous objects are things that are large and have dimensions like a box or a burrito But for particles that can added up by counting we use 2 to represent adding them up For example you might see the following formula don t even bother worrying about it I m making this up E 56273 239 If all of a sudden they want to use the same formula but for continuous objects then all you have to do is change the summation symbol to an integral sign and look at the context to see which is the in nitesimal for the integral I just chose d7quot in this example 25627quot gt c2dr i 15 Choosing Coordinate Axes This is one of the most confusing topics and can lead to wrong answers All of physics is independent of any coordinate system This means that you should get the same answer regardless of what coordinate system you choose You should get the same answer regardless of which direction you call up or down and left or right In particular in ALL problems you could choose ANY coordinate axes and you will end up with the same answer if you are consistent For example the most confusing in the beginning is in kinematics regarding the direction and the sign of gravity Sometimes you ll see it written as g and sometimes as 9 Which is it The answer depends on if you call up positive or down positive Remember that gravity always points down If you call down positive then gravity is g If you call down negative then gravity is g 10 CHAPTER 1 ESSENTIALS Chapter 2 Kinematics 21 Average Velocity and Average Acceleration De nition 21 Given nal position xf and initial position 56 and the change in time At the average velocity is de ned as 56f 56239 At De nition 22 Given nal velocity 12f and initial velocity 1 and the change in time At the average acceleration is de ned as 7 quotOf 0239 At a 22 Instantaneous Velocity and Instantaneous Acceration De nition 23 Instantaneous velocity is de ned as dzc quot12 De nition 24 Instantaneous acceleration is de ned as dv 0 g 23 Four Big Kinematic Equations for CONSTANT ACCELER ATION Memorize please xf zcizvtk a vfzvil at vj2c vi222ad I scf ac vf vit 11 12 CHAPTER 2 KINEMATICS You can use these same equations in each direction because every direction is independent of each other For example you can use these equations separately for both cc and y coordinates Chapter 3 Forces Newton s Laws of Motion 31 Newton s Three Laws of Motion Proposition 31 Newton s 1st Law If there is no net force on an object it stays at rest or moves at constant velocity ie no acceleration Proposition 32 Newton s 2nd Law If a particle is not moving at a constant speed there must be a force acting on it Conversely if the net force is not zero there is an acceleration F 2 ma Note Remember that constant velocity means 0 acceleration Proposition 33 Newton s 3rd Law For every action there is an equal and opposite reaction 32 Free Body Diagrams Types of Forces to Draw There are gravitational forces normal forces frictional forces tensions and spring forces Each object in a system can have its own separate set of axes 321 Gravitational Forces Gravity always points down and the weight of an object is always straight downwards 322 Normal Forces Normal forces is a contact force so anything touching an object will give a normal force 323 Frictional Forces Frictional forces are always point in the opposite direction of the object s motion De nition 34 Kinetic Friction is de ned by F ukN De nition 35 Static Friction is de ned by ansN 13 14 CHAPTER 3 FORCES NEWTON S LAWS OF MOTION 324 Tension Forces Tension forces are always the same along a rope or string 325 Spring Forces Spring forces are from compressing or stretching a spring De nition 36 Spring force is de ned by F ccc Where k is the spring constant and cc is the displacement of how much the spring has been compressed or stretched 33 Static Equilibrium and Dynamics 33 1 Static Equlibrium Static equilibrium means that the system is not moving and the net forces in the cc and y directions are 0 Fnetcc 0 Fnety 0 332 Dynamics Dynamics means that the system is moving and the net forces in the cc and y directions are non zero Fnetcc mam Fnety may Chapter 4 Circular Motion 41 Centripetal Acceleration There is centripetal acceleration when an object is moving around in a circular motion De nition 41 Centripetal acceleration is de ned as a7 E Where v is the velocity of the object in the tangential direction and R is the radius of the circular motion Note Centripetal force is a RESULTANT force This means that aftering summing forces it results in a centripetal force v2 Fnet m R 15 16 CHAPTER 4 CIRCULAR MOTION Chapter 5 Work Energy Energy Conservation 51 Types of Energy 5 1 1 Kinetic Energy This is energy associated with motion De nition 51 Kinetic energy of an object with mass m is de ned as 1 B 2 2m 5 1 2 Potential Energy This is energy associated with location and position such as gravitational potential energy or elastic potential energy De nition 52 Gravitational potential energy is de ned as U9 2 mgh De nition 53 Elastic potential energy from a spring is de ned as 1 Us 2 Elm where k is the spring constant and ac is the displacement that the spring is compressed or stretched 513 Work Work is the transfer of energy from an external force such as friction De nition 54 Work is de ned as WzFszdcose where F is the force applied to an object7 d is the displacement of the object7 and 6 is the angle formed between the force and dispacement 17 18 CHAPTER 5 WORK ENERGY ENERGY CONSERVATION 52 Work Energy Theorem The Work Energy theorem states that the total work done by external forces on an object is the change in kinetic energy Wtotal AK 53 Conservation of Energy Conservation of energy can be stated as Wewt AK AU Note Gravity can be viewed as an external force and can be put in the Wem term7 but it can also be put into the potential energy term instead Chapter 6 Momentum Impulse and Collisions 6 1 Momentum De nition 61 Given mass m and its velocity 17 linear momentum is de ned as 1539 2 m1 Linear momentum can be split into as y and 2 components De nition 62 Given an object s nal momentum 13f and initial momentum 15 impulse is de ned as I A17 17 f 1239 Theorem 63 Impulsemomentum Theorem Change in momentum is equal to the impulse A z f gzFAIS Newton s 2nd Law can be rewritten in terms of momentum mw mdt dt dt FmE 62 Conservation of Momentum Particles in an isolated system meaning there are NO EXTERNAL FORCES have their total momentum conserved Given by 171i 1722 17 172f 63 Collisions Linear momentum is conserved in each direction The following can be applied in both the ac and y directions separately 631 Elastic Collisions Linear momentum is conserved AND kinetic energy is conserved 19 20 CHAPTER 6 MOMENTUM IMPULSE AND COLLISIONS 632 Inelastic Collisions Linear momentum is conserved and kinetic energy is NOT conserved 64 Solving Collision Problems All you have to do to solve these problems is to use conservation of momentum and conservation of energy You can use these equations to solve for Whatever variables you are looking for Chapter 7 Center of Mass The center of mass is the point of an object not necessarily on the object that moves as if all the mass of the object is concentrated at that point De nition 71 The position of the center of mass is de ned as 1 7 CM M mm Z You can split the above vector equation into as y and z equations 1 500M M miivz39 Z 1 yCM M miyi Z 1 ZCM M Z mizz39 239 Velocity and Acceleration of the center of mass can also be de ned as 1 quotUCM H 2 mi Uz39 239 1 CM M Z miaz i 21 22 CHAPTER 7 CENTER OF MASS Chapter 8 Rotational Kinematics Moment of Inertia 8 1 Rotational Analogs Almost all the linear equations have a rotational analog and you can memorize those equations by remembering the following cc gt 6 v gt w a gt a m gt I F gt739 82 Rotational Kinematics Please memorize These are just like the linear versions but the same equations De nition 81 Average Angular Velocity is de ned as A6 w At De nition 82 Average Angular Acceleration is de ned as Aw a At De nition 83 Instantaneous Angular Velocity is de ned as d6 w dt De nition 84 Instantaneous Angular Acceleration is de ned as dw a dt Note To change from revs to rad8 remember that there are 27r in one rev 23 24 CHAPTER 8 ROTATIONAL KINEMATICS MOMENT OF INERTIA 821 Rotational Kinematics under CONSTANT ANGULAR ACCELERATION 1 Qf QZzwtl Eat2 wfzwil oat wj2c wZ222a6 1 822 Linear to Rotational Transformations 8R6 szw azRa 823 Total Acceleration from Moving around in a Circle The tangential acceleration is given by at 2 Rd While the radial acceleration is just centripetal acceleration given by v2 Rea2 E R 2 Ref 17 The total acceleration is given by atom 2 Va l 072 2 RV a2 w4 824 Period from Moving around in a Circle De nition 85 Period T7 is de ned as the circumference divided by the velocity 27TR v T 83 Energy from Moving in Circular Motion De nition 86 Rotational Kinetic Energy is de ned as 1 K I 2 R 2 w Where I is the moment of inertia and w is the angular velocity 84 MOMENT OF INERIA 25 84 Moment of Ineria De nition 87 Moment of inertia for discrete particles is given by I 2 min2 i where mi s are the particles masses and ri s are the particles distances from the aXis of rotation The moment of inertia represents how resistant an object is to rotation De nition 88 Moment of inertia for continuous object is given by r2dm where dm is a very small mass and r is the distance of the small mass from the aXis of rotation Note If an object is made up of many other objects you can add up all the individual moment of inertias total Z Ii i 85 Parallel Axis Theorem Theorem 89 Parallel Axis Theorem states that I CM MD2 where M is the mass of the object and D is the distance from the center of mass to the new rotation axis 86 Rolling without Slipping When an object is not slipping the object is always rotating and always in contact with the surface When an object is slipping it is not necessarily rotating and maybe skidding across the surface De nition 810 Rolling without slipping is de ned by the ability to use the following equations 1 Rw azRa 26 CHAPTER 8 ROTATIONAL KINEMATICS MOMENT OF INERTIA Chapter 9 Torque and Rotational Dynamics 9 1 Torque Torque is analogous to force but it s the angular version Torque is the amount of spin you are applying to make an object turn Remember that counterclockwise is positive and clockwise is negative Torque is always measured with respect to some axis otherwise it means nothing De nition 91 Torque has direction and is de ned as 77rFsin6FgtltFl where F is the displacement from the point of rotation to where force is applied 92 Work in Rotational Motion If an object has been rotated by 6 by a torque 739 then work has been done on the object De nition 92 Work done on an object that has been rotated by 6 by a torque 739 is de ned as W276 93 Rotation Linear Motion Rolling is the most common example of both rotation and linear motion Rolling can be modeled as a combination of pure linear motion and pure rotational motion 931 Kinetic Energy of Rolling Kinetic Energy of Rolling is the sum of the linear translational energy of its center of mass and the rotational kinetic energy of its center of mass 1 1 K 5 CMLU2 27 28 CHAPTER 9 TORQUE AND ROTATIONAL DYNAMICS Chapter 10 Angular Momentum 101 Angular Momentum De nition 101 Angular momentum is de ned as E 13 Where I is the moment of inertia and w is the angular velocity Note that angular momentum is a vector Angular momentum can also be de ned as follows E 2 FX 15 lrllmvlsin l Where F is the displacement from the point of rotation to Where the momentum acts 102 Conservation of Angular Momentum Conservation of angular momentum occurs When there are NO EXTERNAL TORQUES This means that both magnitude and direction of angular momentum is conserved It is given by 5225 103 When there are External Forces When there are external forces you normally have Fnet 65 13 So for angular momentum you would have stotal Tnet dt Note Torque and angular momentum are measured about the same aXis of rotation 29 Index A angular momentum 29 average acceleration 11 average angular acceleration 23 average angular velocity 23 average velocity 11 C center of mass 21 circular motion 15 collisions 19 conservation of angular momentum 29 conservation of energy 18 conservation of momentum 19 coordinate axes 9 cross product 8 D determinant 8 dot product 7 dynamics 14 E elastic collisions 19 elastic potential energy 17 energy 17 external force 17 F free body diagrams 13 frictional forces 13 G gravitational forces 13 gravitational potential energy 17 I impulse 19 impulse momentum theorem 19 inelastic collisions 20 instantaneous acceleration 11 instantaneous angular acceleration 23 instantaneous angular velocity 23 instantaneous velocity 11 K kinetic energy 17 kinetic friction 13 L linear momentum 19 M moment of inertia 25 momentum 19 N Newton s Three Laws of Motion 13 normal forces 13 P parallel aXis theorem 25 period 24 potential energy 17 R right hand rule 8 rolling Without slipping 25 rotational kinematics 23 rotational kinetic energy 24 S scalars 7 spring constant 14 spring force 14 static equilibrium 14 static friction 13 T tension forces 14 torque 27 V vector 7 30 INDEX W work 17 work energy theorem 18 31