PHYS2205. General Physics Classical Mechanics (including waves) Prof. Piilonen.

Transcription

1 PHYS2205 General Physics Classical Mechanics (including waves) 1. Register with your personal access code Prof. Piilonen Log in and join course MPPIILONEN70022 Log in to Scholar Choose this course Read the syllabus

2 Units We will use SI (Système Internationale) units: seconds, meters, kilograms, etc. We will avoid British / Imperial units and unit conversions as much as possible Unit prefixes (powers of ten) are used so that the numerical value is between and 1000: nano-, micro-, milli-, centi-, kilo-, mega-, giga-, etc. [see back of textbook] Uncertainty / Significant Figures Uncertainty ( precision and accuracy ) will be confined to the laboratory (PHYS 2215). For homework, quizzes, and tests, three significant figures should be used for the final answer unless you re told otherwise (but you should retain more significant figures for intermediate values). 20 kilograms vs 20,000 grams 0.3 millimeters vs meters Scalars A scalar has a value and a unit but no direction. temperature, weight, frequency,... Equal values of a scalar quantity can be connected by a contour. Vectors A vector has a value and a unit and a direction. displacement, velocity, force,... A vector is drawn as straight-line arrow, usually with a length proportional to the vector s value. wind velocity a contour of constant elevation on a map Vectors A vector is drawn as straight-line arrow. The value of each vector is indicated by its color instead of its length. Vectors These arrows are not vectors because they are crooked.

3 Adding Two Vectors Geometrically Place the tail of vector B at the head of vector A. The sum, vector C, points from the tail of A to the head of B. Adding Two Vectors Geometrically or... place the tail of vector A at the head of vector B. The sum, vector C, points from the tail of B to the head of A. Addition of vectors is commutative (i.e., order of A and B is immaterial). Adding Two Vectors Geometrically Adding Three Vectors Geometrically Place the tails of vectors A and B together, then draw a parallelogram. The sum, vector C, points from the common tail of the parallelogram to the opposite corner. two-step addition { D = A + B R = D + C E = B + C R = A + E { one-step addition R = A + B + C R = B + A + C What is the direction of the vector A + B + C? What is the direction of the vector A + B + C? C B R = A + B + C A

4 If angles are measured CCW from the +x axis, use cosine for x component and sine for y component. If angles are measured CCW from the +x axis, use cosine for x component and sine for y component. Bx = B cos θ Cx = C cos θ Ax = A cos θ By = B sin θ Cy = C sin θ Ay = A sin θ because 90 < θ < 180 If positive angles are measured in the interior of a right triangle, use cosine for adjacent and sine for opposite, then decide if positive or negative. Dx = +D cos α Dy = D sin α 0 < α < 90 by construction, so we put in the + and signs for this particular triangle. Adding Two Vectors Componentwise Extract the components of vectors A and B. Add the x components of A and B to get the x component of C. Add the y components of A and B to get the y component of C. Ax = A cos 60 Ay = A sin 60 Bx = B cos 0 = B By = B sin 0 = 0 Cx = Ax + Bx = A cos 60 + B Cy = Ay + By = A sin because 180 < θ < 270 If positive angles are measured in the interior of a right triangle, use cosine for adjacent and sine for opposite, then decide if positive or negative. Ex = +E sin β Ey = +E cos β 0 < β < 90 by construction, so we put in two + signs for this particular triangle. What are the components of the vector A

5 What are the components of the vector A Use angles measured CCW from +x axis. What are the components of the vector A Use interior angle of right triangle (as illustrated). Ax = (100) cos 30 = 87 Ax = +(100) cos 30 = 87 Bx = (80) cos 120 = 40 Bx = (80) cos 30 = 40 Ay = (100) sin 30 = 50 Ay = +(100) sin 30 = 50 By = (80) sin 120 = 69 By = +(80) sin 30 = 69 Cy = (40) sin 233 = 32 Cy = (40) sin 53 = 32 Cx = (40) cos 53 = 24 Cx = (40) cos 233 = 24 Rx = = 23 Ry = = 87 and Vector Magnitude and Direction Given the components of a vector, you can get its magnitude ( size ) and direction ( angle ): = B = C Bx2 + By2 " By θ = tan 1 Bx R = 90 Unit Vectors A unit vector points along a particular direction and has a value of 1 with no units (i.e., is dimensionless). y is a unit vector parallel to the y axis. Rx = 23 Ry = 87 A and What is the magnitude and direction of the vector A y B Ry = = 87 Cx2 + Cy2 " Cy θ = tan 1 Cx C = R and What is the magnitude and direction of A Components of the sum vector were obtained earlier. Rx = = 23 x x is a unit vector parallel to the x axis. θ = tan 1 (87/23) = 75 (CCW from + x axis) for a generic vector. Use ˆ for a unit vector. Use

6 Unit Vectors We combine the components with the unit vectors to express a vector. A y ŷ what is the vector from Manhattan to Lincoln using unit-vector notation? x measured relative to North ŷ ˆx A x ˆx y A = A x ˆx + A y ŷ A =(A x,a y ) { Two ways to write the vector A. Don t combine with A x A y what is the vector from Manhattan to Lincoln using unit-vector notation? measured relative to North what is the distance between Manhattan and Lincoln, to the nearest km? measured relative to North x y what is the direction from Manhattan to Lincoln, to the nearest degree? R = 186 ˆx 34.3ŷ (i.e., a little west of due north) what is the distance between Manhattan and Lincoln, to the nearest km? measured relative to North what is the direction from Manhattan to Lincoln, to the nearest degree? 189 km 10º (i.e., 10º west of north)