When a slice of buttered toast is accidentally pushed over the edge of a counter, it rotates as it falls. If the distance to the oor is 54cm and for rotation less than 1 rev, what are the (a) smallest and (b) largest angular speeds that cause the toast to hit and then topple to be butter-side down? Assume free-fall acceleration to be equal to 9.81 m/s2. n? n n? I eTextbook and Media Assistance Used m Assistance Used The falling is the type of constant-acceleration motion. You need the time of falling. Through what angle must the toast turn while falling so that after it rst hits it topples over to be butter side down? Angular speed is the ratio of the angle turned to the time interval of turning. Key Ideas 0 To describe the rotation ofa rigid body about a xed axis, called the rotation axis, we assume a reference line is fixed in the body, perpendicular to that axis and rotating with the body. We measure the angular position 6 ofthis line relative to a fixed direction. When 9 is measured in radians. S , 6 = (radian measure), r where sis the arc length ofa circular path of radius r and angle 61 - Radian measure is related to angle measure in revolutions and degrees by 1 rev 2 360 = 2n rad. - A body that rotates about a rotation axis, changing its angular position from 91 to 02, undergoes an angular displacement A9 = 02 01, where A9 is positive for counterclockwise rotation and negative for clockwise rotationi 0 Ifa body rotates through an angular displacement A9 in a time interval At, its average angular velocity wavg is A0 wavg = E' The [instantaneous] angular velocity u) of the body is d6 (0 = . (it Both wavg and w are vectors, with directions given by a righthand rule. They are positive for counterclockwise rotation and negative for clockwise rotation. The magnitude of the body's angular velocity is the angular speed. - Ifthe angular velocity ofa body changes from (1)1 to )2 in a time interval At = t2 , t1. the average angular acceleration (1an of the body is a); an A0) a = = i avg 22 [1 At The [instantaneous] angular acceleration crofthe body is dw a = . dt Both (1an and a are vectors