PHY 2049L Lecture Notes - Lecture 6: Circular Motion, Angular Velocity, Normal Force

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28 Jul 2017

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Displacement of position vector r = r_f - r_i = (x_f - x_i)i + (y_f - y_i)j. Velocity of position vector v = dr/dt = (dx/dt)i + (dy/dt)j. Acceleration of position vector a_a(cid:448) = (cid:894)(cid:448)(cid:1031) - (cid:448)(cid:1030)(cid:895)/(cid:894)t(cid:1031) - t(cid:1030)(cid:895) = (cid:448)/ t a = dv/dt. Two-dimensional motion w/ constant acceleration (cid:448) = (cid:448)(cid:1029) + at r = r(cid:1029) + (cid:448)(cid:1029)t + at. Velocity as function of position (cid:894)(cid:448)_(cid:454)(cid:895) = (cid:894)(cid:448)_(cid:454)(cid:895)(cid:1029) + 2(cid:894)a_(cid:454)(cid:895)(cid:454) (cid:894)(cid:448)_(cid:455)(cid:895) = (cid:894)(cid:448)_(cid:455)(cid:895)(cid:1029) + 2(cid:894)a_(cid:455)(cid:895)(cid:455) When object is given initial velocity and follows path determined entirely by effect of gravitational force. Projectile motion formulae (cid:448)_(cid:454) = (cid:448)(cid:1029)(cid:272)os(cid:894)a(cid:1029)(cid:895) (cid:448)_(cid:455) = (cid:448)(cid:1029)si(cid:374)(cid:894)a(cid:1029)(cid:895) - gt (cid:454) = (cid:454)(cid:1029) + (cid:448)(cid:1029)(cid:272)os(cid:894)a(cid:1029)t(cid:895) (cid:455) = (cid:455)(cid:1029) + (cid:448)(cid:1029)si(cid:374)(cid:894)a(cid:1029)t(cid:895) - gt . Velocity of magnitude is constant but direction continuously changes; Velocity at each point is tangent to circle and acceleration is directed toward center of circle. Angle for which arc length of a circle of radius r is equal to radius of circle.

Related Questions

A particle P moves with constant angular speed ω around a circle whose center is at the origin and whose radius is R.The particle is said to be in a uniform circular motion. Assume that the motion is counterclockwise and that the particle is at the point (R, 0) when t = 0.The position vector at time t ≥ 0 is r(t) = R cos ωt i + R sin ωt j.

(a) Find the velocity vector v and show that v · r = 0. Conclude that v is tangential to the circle and points in the direction of the motion.

(b) Show that the speed of the particle is constant ωR. The period T of the particle is the time required for one complete revolution. Conclude T = 2π/ω.

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(A) Show that the position of a particle on a circle of radius Rwith center at the origin (x=0,y=0) is given in unit vectors byr=cosÎ¸i+sinÎ¸j, where Î¸ is the angle of the position vector with thex axis.

B) If the particle moves with constant speed v and period T,starting on the x axis at t=0, find an expression for Î¸ in terms ofthe time and T.

C) Differentiate the position vector twice with respect to timeto find the acceleration and show that it is the centripetalacceleration, whose magnitude is given by equation (a=(v^2)/r) andwhose direction is toward the center of the circle. (Not that theunit vectors i and j are constants, independent of time.)

The coordinates of an object moving in the xy plane vary with timeaccording to x= -(5.00 m) sin(wt) and y= (4.00 m) - (5.00m)cos(wt), where w is a constant and t is in seconds.
a) Determine the components of velocity and components ofaccelerationof the object at t=0.
b) Write expressions for the position vector, the velocity vector,and the acceleration vector of the object at any time t>0.
c) Descibe the path of the object in an xy plot.

PHY 2049L Lecture Notes - Lecture 6: Circular Motion, Angular Velocity, Normal Force

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