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Transcript
Two-Dimensional Motion Projectile Motion Periodic Motion Projectile Moion Vx Vx Vy Vx Vy Vx = constant Vy = varying Vy Vx Vx Vy Vx Vy Formulas: Vx = constant therefore, Vx = d/t Vy = varying therefore, acceleration vf = vi + at vf2 = vi2 + 2ad d = vi + 1/2at2 Vy Projectile Motion vi q vx vy Vy = sinq(vi) Vx = cosq(vi) Vy controls how long it’s in the air and how high it goes Vx controls how far it goes Projectile Motion “Range formula” Remember!!!!! vi yi vi is the velocity at an angle and the sin2q is the sine of 2 x q R = vi2 sin2q/g Range formula works only when yi = yf yf Projectile Motion “Range formula” R = vi2 sin2q/g vi yi If vi = 34 m/s and q is 41o then, R = (34 m/s)2 sin82o/9.8 m/s2 R = 1160 m2/s2 (0.99)/9.8 m/s2 R = 120 m yf Projectile Motion “Range formula” Note that if q becomes the complement of 41o, that is, q is now 49o, then, vi q vi = 34 m/s and q is 49o then, R = (34 m/s)2 sin98o/9.8 m/s2 R = 1160 m2/s2 (0.99)/9.8 m/s2 R = 120 m So, both 41o and 49o yield “R” Projectile Motion “Range formula” OR, vi yi q vy If vi = 34 m/s and q is 41o then, vx vy = sin41o(34m/s) = 22m/s, and t = vfy - viy/g = -22m/s - (22m/s)/-9.8m/s2 = 4.5 s vx = cos41o(34 m/s) = 26 m/s, and dx = vx(t) = 26m/s (4.5 s) = 120 m yf Circular Motion When an object travels about a given point at a set distance it is said to be in circular motion Cause of Circular Motion 1st Law…an object in motion stays in motion, in a straight line, at a constant speed unless acted on by an outside force. 2nd Law…an outside force causes an object to accelerate…a= F/m THEREFORE, circular motion is caused by a force that causes an object to travel contrary to its inertial path Circular Motion Analysis v1 v2 r q r Circular Motion Analysis v1 v1 q v2 r q r v2 Dv = v2 - v1 or Dv = v2 + (-v1) (-v1) = the opposite of v1 v1 (-v1) v1 v1 q v2 r r 0 v2 Dv = v2 - v1 or Dv = v2 + (-v1) (-v1) = the opposite of v1 v1 Dv v2 q (-v1) (-v1) Note how Dv is directed toward the center of the circle v1 Dl r q v1 q v2 r v2 Dv v2 q (-v1) Because the two triangles are similar, the angles are equal and the ratio of the sides are proportional v1 v1 q Dl r q v2 v2 r v2 Dv q (-v1) Therefore, Dv/v ~ Dl/r and Dv = vDl/r now, if a = Dv/t, and Dv = vDl/r then, a = vDl/rt, since v = Dl/t THEN, a = v2/r Centripetal Acceleration ac = v2/r now, v = d/t and, d = c = 2pr then, v = 2pr/t and, ac = (2pr/t)2/r or, ac = 4p2 r2/t2/r or, ac = 4p2r/T2 The 2nd Law and Centripetal Acceleration vt Fc F = ma ac ac = v2/r = 4p2r/T2 therefore, Fc = mv2/r or, Fc = m4p2r/T2 Simple Harmonic Motion or S.H.M. Simple Harmonic motion is motion that has force and acceleration always directed toward the equilibrium position and has its maximum values when displacement is maximum. Velocity is maximum at the equilibrium position and zero at maximum displacement Pendulum motion, oscillating springs (objects), and elastic objects are examples Simple Harmonic Motion F = max a = max v=0 F = less a = less v = greater F=0 a=0 v = max F = greater a = greater v = less F = max a = max v=0 Force acceleration Pendulum Motion Note that FT (the accelerating force is a component of the weight of the bob that is parallel to motion (tangent to the path at that point). Fw FT Pendulum Motion Fw FT Note that as the arc becomes less so does the FT, therefore the force and resulting acceleration also becomes less as the “bob” approaches the equilibrium position. Pendulum Motion ac = 4p2 r/T2 ac = g and r = l g = 4p2 l/T2 T2 = 4p2 l/g Fw T = 2p FT l/g Oscillating Elastic Objects Fe = max Fe = less Fe = less Fe = max a = max a = less a = less a = max a and F =0 Note that no part of Fw is in the direction on Motion, or FT There, F and a is zero!!! FT Fw