By differentiating the equation b=tx n, derive the equation b' = -tn. (2) calculate the frenet apparatus (t, k, n, b, t) of the following curve a(t) = (1/2) (sint + cost), (1/2)e' (sint - cost), e) (3) check whether the following bases are positive: (a) the basis {(1,3), (4,2)} in rp (b) the basis {(1,3,5), (2,3,7), (4,8,3)} in r. (4) a circular disk of radius 1 in the (x, y) plane rolls without slipping along the r axis. the figure described by a point on the circumference of the disk is called a cycloid (figure 1-7 in the text). write the equation that describes a parametrized curve a: r → r², whose image is the cycloid. determine its singular points. (5) let (0 ) r2 he given by

Date chapter motion and speed review part a. vocabulary review directions: write the terms that match each description below in thes complete the sentence in number 13. ich each description below in the spaces provided. the boxed letters will 1. equal forces acting in opposite directions (2 words) 2. occurs when an object's velocity decreases (2 words) 3. sum of all forces acting on an object (2 words) 4. total distance divided by total time (2 words) 5. name for the slant of a line on a graph 6. distance divided by time 7. a push or a pull exerted on a body or object 8. distance and direction of an object from its starting point 9. tendency to resist change in motion 10. measure of how far an object has moved 11. speed and direction of motion 12. rate of change of position at a specific point in time (2 words) 13. the rate of change in velocity is called motion and spe

Part f - example: finding two forces (part i) two dimensional dynamics often involves solving for two unknown quantities in two separate equations describing the total force. the block in (figure 1) has a mass m=10kg and is being pulled by a force f on a table with coefficient of static friction îľs=0.3. four forces act on it: the applied force f (directed î¸=30â above the horizontal). the force of gravity fg=mg (directly down, where g=9.8m/s2). the normal force n (directly up). the force of static friction fs (directly left, opposing any potential motion). if we want to find the size of the force necessary to just barely overcome static friction (in which case fs=îľsn), we use the condition that the sum of the forces in both directions must be 0. using some basic trigonometry, we can write this condition out for the forces in both the horizontal and vertical directions, respectively, as: fcosî¸â’îľsn=0 fsinî¸+nâ’mg=0 in order to find the magnitude of force f, we have to solve a system of two equations with both f and the normal force n unknown. use the methods we have learned to find an expression for f in terms of m, g, î¸, and îľs (no n).

Alinearly polarized electromagnetic wave has an average intensity of 196 w/m^2. this wave is directed towards two ideal polarizers (in real polarizers, transmission is also effected by reflection and absorption). polarizer a is oriented with its transmission axis at an angle of θ_1=20.8∘ with the incident electric field. polarizer b has its axis at an angle of θ_2=63.0∘ with the incident electric field. what is the average intensity of the wave after it passes through polarizer a? what is the average intensity of the wave after it passes through polarizer b? suppose that the two polarizers a and b are interchanged. what would the average intensity be after passing through both polarizers?