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Unformatted text preview: Physics 2A
Olga Dudko UCSD Physics Lecture 20
Today: Relating Rotational and Linear motion Rotational vectors and Right Hand Rule Equations for constant angular acceleration
constant-acceleration equations apply with the proper substitutions:
Equation If angular acceleration is constant, then all our Missing Quantity v = v 0 + at
1 x = (v 0 + v)t 2 1 x = v 0 t + at 2 2 = 0 +
+ )t 1 = ( 2 =
2 1 2 t 0t + 2
2 0 v = v + 2a x 2 2 0 = +2 t Example Rotational Variables A wheel has a constant angular acceleration of 3.0rad/s2. During a certain 4.0 sec interval, you measured that the wheel had an angular displacement of 120 radians. If the wheel started at rest, how long had it been in motion at the start of the 4 second interval?
4 sec, 120rad t=? Solution First, you must define a coordinate system. Let's say that the wheel starts moving counterclockwise in the positive direction. Rotational Variables
Solution (cont'd) The quantities we know: (it starts from rest) o = 0 = +3.0rad/s2 We also know that in a 4sec. span it went through an angle of 120rad. <-- don't know <-- don't know <-- finding
+ )t 1 2 t 2 1 = ( 2 =
2 0t + = 2 0 +2 4 sec, 120rad t=? t We can use the 3rd equation to find the angular velocity 0 at the beginning of that time span. Rotational Variables Use:
0 = 1 2 t 2 1 2 t 0t + 2
0 t = = t 1 2 2 t 4 sec, 120rad =2 120 rad - 1 3 rad 2 (4s)2 2 s = 24 rad s 0 = 4s with an angular velocity of 24rad/s. We have a new known: Thus, the wheel started out the 4 sec. interval
= 24rad/s. 4r ad Solution (cont'd) t=? /s Answer Since
0 Rotational Variables
is missing we can use:
4 sec, 120rad = 0 +
+ )t 1 2 t 2 1 = ( 2 = 0t + 24 rad/s t= = 8 sec 2 3 rad/s =0 t = 8 sec 2 / = 2 0 +2 Relating linear and rotational variables Let's say you were traveling on the edge of a merry-go-round of radius, r. You want to know your distance travelled, s, if you have rotated through an angular displacement, . s = r [
r in radians]
s The bigger the radius, the more distance you have covered per angular displacement. Relating Variables Now, let's say your angular velocity is and you want to know how fast you would travel linearly if you let go of the merry-go-round.
s displacement = vt = t time vt = r t =r t = vt = r where vt is known as tangential speed. Relating Variables What is the relation between acceleration (a) and angular acceleration ( ) ? First, there is centripetal acceleration, ac:
ac = v = r
2 t (r ) 2 r ac = 2r acceleration, ar. Radial acceleration component tells us that the body is traveling in a circular type motion (<=> the velocity vector is changing direction). This acceleration is also known as radial Relating Variables The second type of acceleration is tangential acceleration, at:
dv d at = t = d(r ) = r dt dt = dt at = r This tangential acceleration component tells us if the body is changing its angular speed (<=> is changing magnitude).
at ar ar a at a = at2 + ar2 a= 2 2 r + 4 2 r Clicker Question
Andrea and Chuck are riding on a merry-go-round. Andrea rides on a horse at the outer rim of the circular platform, twice as far from the center of the circular platform as Chuck, who rides on an inner horse. When the merry-go-round is rotating at a constant angular speed, Andrea's tangential speed is: A) a quarter of Chuck's. B) half of Chuck's. C) the same as Chuck's. D) twice Chuck's. E) four times Chuck's. Right Hand Rule Angular displacement angular acceleration Direction is defined by the Right Hand Rule: Grasp axis of rotation with your right hand. Wrap your fingers in the direction of rotation. Your thumb points in the direction of . , angular velocity , and are all vector quantities. We already have equations for the magnitude. Right hand rule This disk rotates clockwise => the
angular velocity vector points... ...into the board. This disk rotates counterclockwise =>
the angular velocity vector points... ...out of the board. ! Remember to use your right hand when performing the right hand rule. Another way to multiply vectors is the cross product:
r r r C=A B Cross Product Magnitude of C is given by:
where r r C = A B sin is the angle between vectors A and B. by the Right Hand Rule. Direction of C is determined
http://physics.syr.e du/ C is to both A and B. A cross product measures how perpendicular two vectors (A and B) are. If A || B => their cross product is zero. Cross Product Thus, for unit vectors: ^ Also, for and : i
^ ^ ^ ^ ^ ^ ^ ^ k ^ j 0 j = k k =?
^ ^ ^ ^ ^ i j = k j i= k r r For 2D vectors A and B: A = Ax^ + Ay j^ B = Bx^ + By j^ i i the cross product is r r ^ ^ ^ ^ A B = ( Ax i + Ay j ) (Bxi + By j )
= Ay Bx ( j i ) + Ax By (i = ( Ax By - Ay Bx )k
^ ^ ^ ^ j) ^ Warning! The cross product is not commutative:
r r r r A B B A In fact: r r r r A B = -(B A ) For Next Time: Read Chapter 12 Work on homework for Chapter 12 ...
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