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Honors Physics 1 Class 04 Fall 2013 Vectors Non-Cartesian coordinate systems Motion in multiple dimensions Uniform circular motion Applications 1 Activity – Motion with constraints Frictionless pulley and table. Only net external force=gravity on mass 2. Rope massless and fixed length. Find acceleration of mass 1. 3rd Law: Fs 2 F2 s ; Fs1 F1s ; F1s F2 s T Constraint: The velocity and acceleration of M1 is the same as M2 Do "Free Body", dude! Mass 1, Mass 2 T M1a M 2 a M 2 g T M 2 g M1a 2 Vectors r ( x, y, z ) rxiˆ ry ˆj rz kˆ rxiˆ ry ˆj rz kˆ dx ˆ dy ˆ dz ˆ dr ˆ ˆ ˆ v vxi v y j vz k i j k dt dt dt dt Note: differentiating vectors requires care. ˆ dx di The derivative of xiˆ is really: iˆ x . dt dt 2 2 2 2 d x d y d z d r ˆ ˆ ˆ ˆ ˆ ˆ a a xi a y j a z k 2 i 2 j 2 k 2 dt dt dt dt v adt ( v0 at for constant a ) Examples: Projectile motion; Uniform circular motion. 3 Cartesian coordinates iˆ, ˆj , kˆ form a right-handed coordinate system iˆ ˆj kˆ; ˆj kˆ iˆ; kˆ iˆ ˆj r xiˆ yjˆ zkˆ dr dxiˆ dyjˆ dzkˆ dV dxdydz 4 Cylindrical coordinates dV d d dz ds d ˆ dˆ dzzˆ x cos , y sin , zz ( x 2 y 2 )1/2 , tan y / x uˆ uˆ 0; uˆ uˆ z 0; uˆ uˆ z 0 uˆ uˆ uˆ z ; uˆ uˆ z uˆ ; uˆ z uˆ uˆ uˆ 1 uˆ uˆ z z 5 5 Spherical coordinates x r sin cos ; y r sin sin ; z r cos r x y z ; tan y / x; tan 2 2 2 x2 y 2 z dV r 2 sin drd d ds drrˆ rdˆ r sin dˆ 1 1 uˆ uˆ r r r sin uˆr uˆ uˆ ;... uˆr 6 6 Exponentials and complex numbers Complex numbers and the Euler relation are sometimes useful tools for solving problems. We start with the definition i 1. And the representation of a complex number, z a ib on a plane where a is the value on the x-axis and b is the y-axis so z = a b 2 2 1/2 . By comparing series representations of sin , cos , and ei we know that: ei cos i sin ei e i ei e i and cos ;sin 2 2i Discuss powers, square roots, derivatives. 7 Newton’s Laws in 3d form F Fi ma Fx ma x ; Fy ma y Because x and y components are at right angles, the equations can be solved independently. Choice of coordinate system is a powerful tool. 8 Motion Example: Projectile trajectory A tennis ball is hit at 50 mps at an angle of 5 degrees above the horizontal. The initial height is 2 m. Neglecting air drag, how far does the ball go before hitting the ground? Choose +y to be up and x to be horizontal. y0 2; v0 x v0 cos ; v0 y v0 sin ; a x 0; a y 9.8 1 y y0 v0 y t a y t 2 2 Solve for time at which y=0. Use x x0 v0 xt to find distance. 9 Motion Example: Choosing coordinates well is important A tennis ball is hit at 50 mps at an angle of 5 degrees above the horizontal. The initial height is 2 m. Neglecting air drag, how far does the ball go before hitting the ground? Choose +x to be in the direction the ball starts at. Choose +y to be at right angles to that. Choose the origin to be the starting point. y0 0; x0 0; y f 2 / cos v0 x v0 ; v0 y 0; a0 x 9.8sin ; a0 y 9.8cos (The x that you first solve for is not the landing point.) 10 Activity: Uniform circular motion Let's describe the position, velocity, and acceleration of an object moving in uniform circular motion. (Constant angular speed . Constant radius .) x cos cos t ; y sin t r xiˆ yjˆ iˆ cos t ˆj sin t v ? a ? a ? Which way does a point? Which way does v point? v ? 11 Activity: Application The Spinning Terror ride The spinning terror is a large vertical drum which spins so fast that everyone stays pinned to the wall when the floor drops out. For a typical ride the radius of the drum is 2 m. What is the minimum angular velocity if the coefficient of friction between the patron and the wall is 0.3? 12 Application example: Falling through a viscous fluid Assume that the density of the fluid is very small compared to the density of the falling object. (e.g – a human body in air) Assume that the body falls under the action of constant gravity and drag force only. Assume that the drag force is linear in speed: FD Cv Is there a terminal velocity? If there is, find the terminal velocity. 13 Application example: Mass on a spring Equilibrium position x 0; Starting point=x0 F k x Write the F ma equation. Assume a solution of the form: x(t ) Ae t . See what conditions have to be met by A and to solve the relation F ma and satisfy initial conditions. 14