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AP Physics C Math Review Vector Components • All vectors can be broken into their individual x & y components. • Typically - x r cos y r sin • However, if you have an inclined plane this will not be the case. Make sure you determine the x & y components with the angles in your diagrams. Vector Addition • Break vectors into x & y components • Add the components together • Report answer using magnitude and direction or components. • î & ĵ are used to represent x and y • You will use vector addition in kinematics, with forces, momentum, electric and magnetic field, and anything else that is a vector. • Energy and electric potential are not vectors Vector Multiplication Dot Product • If we want the parallel components of two vectors we use the dot product. • We use the cosine of the angle between the two vectors. Work is shown below… W F d Fd cos • Dot product is used with work, power (P=Fv), electric and magnetic flux, and Ampere’s Law. Vector Multiplication Cross Product • If we want the perpendicular components of two vectors we use the dot product. • We use the sine of the angle between the two vectors. Torque is shown below… r F rF sin • Cross product is used with torque, angular momentum, magnetic force (for both a charge and a current) • Direction is given by right hand rule. Index finger points in the direction of the first variable, middle finger with the second variable and the thumb is the direction of the result. Differential Equations • Differential equations are used to solve for objects that experience air resistance, harmonic motion, RL circuits, RC circuits, LC circuits Differential Equations Retarding Force (air resistance) Moving Forward FD kv dv m a m kv dt Falling Object F m g kv dv ma m m g kv dt • Use Newton’s Second Law and identify the forces on the object. • Separate variables and integrate to solve. Be sure to include limits of integration • You will end with something like e-kx or A(1 - e-kx ) Harmonic Motion • The motion is periodic – it repeats itself and can be modeled as a sine function • Springs and LC circuits all have the same form. • Pendulums are also harmonic Spring F m a kx d 2x k x0 2 dt m k m LC Circuit dU di q dq Li 0 dt dt C dt d 2q 1 q0 2 dt LC 1 LC RC Circuits Charging • Use Kirchoff’s Loop rule to find voltage across each component. Set the current to dq/dt. • Separate the variables and solve for q. ε C R q iR 0 C dq q R dt C q C (1 e t / RC ) dq t / RC i (t ) e dt R RL Circuits Rise of Current • Use the same approach as the RC circuit – the difference is that we look at current instead of charge. di iR L 0 dt di L iR dt i R (1 e Rt / L ) Integration for AP Physics • There are many cases where you will be asked to integrate on the AP Physics exam or when you are asked to find the area under a curve. • The next few slides will deal with the more difficult integration that you may encounter. • There are some common themes among: Center of Mass, Rotational Inertia, Electric Field and Electric Potential Integration for AP Physics For the situations listed in the last slide there are some common strategies to approach the problems. 1. Identify symmetry and choose your axes so that you integrate along a line of symmetry 2. Use the mass/charge density and break your object into pieces that fit the shape. Integration for AP Physics Linear • Break the long, thin rod into small pieces dx • Use the mass/charge density to find dm or dq in terms of dx. • Substitute in order to get a single variable in your integral • Plug values in, set limits of integration and go Integration for AP Physics Two Dimensional • Break the area into shapes – either rings or rectangles • Use the mass/charge density to solve for dm or dq in your ring • Substitute in order to get a single variable in your integral • Plug values in, set limits of integration and go Integration for AP Physics Three Dimensional • Break the area into shapes – a sphere breaks into spherical shells. A cylinder breaks into cylindrical shells. • Use the mass/charge density to solve for dm or dq in your shape • Substitute in order to get a single variable in your integral • Plug values in, set limits of integration and go Integration Other Notes • E-fields are vectors and you need to keep the direction in mind as you solve • Center of Mass, Rotational Inertia and Potential are scalar quantities