Machinery's Handbook, 31st Edition
184 Velocity and Acceleration Table 2 can be used to obtain angular velocity in radians per second for all whole num- bers of revolutions per minute from 1 to 239. Table 2. Angular Velocity in Revolutions per Minute Converted to Radians per Second RPM Angular Velocity in Radians per Second (rad/s) 0 1 2 3 4 5 6 7 8 9 0 0.00 0.10 0.21 0.31 0.42 0.52 0.63 0.73 0.84 0.94 10 1.05 1.15 1.26 1.36 1.47 1.57 1.67 1.78 1.88 1.99 20 2.09 2.20 2.30 2.41 2.51 2.62 2.72 2.83 2.93 3.04 30 3.14 3.25 3.35 3.46 3.56 3.66 3.77 3.87 3.98 4.08 40 4.19 4.29 4.40 4.50 4.61 4.71 4.82 4.92 5.03 5.13 50 5.24 5.34 5.44 5.55 5.65 5.76 5.86 5.97 6.07 6.18 60 6.28 6.39 6.49 6.60 6.70 6.81 6.91 7.02 7.12 7.23 70 7.33 7.43 7.54 7.64 7.75 7.85 7.96 8.06 8.17 8.27 80 8.38 8.48 8.59 8.69 8.80 8.90 9.01 9.11 9.21 9.32 90 9.42 9.53 9.63 9.74 9.84 9.95 10.05 10.16 10.26 10.37 100 10.47 10.58 10.68 10.79 10.89 11.00 11.10 11.20 11.31 11.41 110 11.52 11.62 11.73 11.83 11.94 12.04 12.15 12.25 12.36 12.46 120 12.57 12.67 12.78 12.88 12.98 13.09 13.19 13.30 13.40 13.51 130 13.61 13.72 13.82 13.93 14.03 14.14 14.24 14.35 14.45 14.56 140 14.66 14.76 14.87 14.97 15.08 15.18 15.29 15.39 15.50 15.60 150 15.71 15.81 15.92 16.02 16.13 16.23 16.34 16.44 16.55 16.65 160 16.75 16.86 16.96 17.07 17.17 17.28 17.38 17.49 17.59 17.70 170 17.80 17.91 18.01 18.12 18.22 18.33 18.43 18.53 18.64 18.74 180 18.85 18.95 19.06 19.16 19.27 19.37 19.48 19.58 19.69 19.79 190 19.90 20.00 20.11 20.21 20.32 20.42 20.52 20.63 20.73 20.84 200 20.94 21.05 21.15 21.26 21.36 21.47 21.57 21.68 21.78 21.89 210 21.99 22.10 22.20 22.30 22.41 22.51 22.62 22.72 22.83 22.93 220 23.04 23.14 23.25 23.35 23.46 23.56 23.67 23.77 23.88 23.98 230 24.09 24.19 24.29 24.40 24.50 24.61 24.71 24.82 24.92 25.03 Example: To find the angular velocity in radians per second of a flywheel making 97 rev olutions per minute, locate 90 in the left-hand column and 7 at the top of the columns; at the intersection of the two lines, the angular velocity is read off as equal to 10.16 radians per second. Linear Velocity of Points on a Rotating Body.— The linear velocity, v , of any point on a rotating body expressed in feet per second may be found by multiplying the angular veloc ity of the body in radians per second, ω , by the radius, r , in feet from the center of rotation to the point: (2) The metric SI units are ν = meters per second; ω = radians per second, r = meters. Rotary Motion with Constant Acceleration.— The relations among angle of rotation, an- gular velocity, and time for rotation with constant or uniform acceleration are given in the accompanying Table 3. In these formulas, the acceleration is assumed to be in the same direction as the initial angular velocity; hence, if the acceleration in a particular problem should happen to be in a direction opposite that of the initial angular velocity, then α should be replaced by −α . Thus, for example, the formula ω f = ω o + α t becomes ω f = ω o − α t when α and ω o are opposite in direction. Linear Acceleration of a Point on a Rotating Body: A point, P , on a body rotating about a fixed axis has a linear acceleration a that is the resultant of two component accelerations. The first component is the centripetal, or normal, acceleration, which is directed from the point P toward the axis of rotation; its magnitude is r ω 2 , where r is the radius from the axis to the point P and ω is the angular velocity of the body at the time acceleration a is to be determined. The second component of a is the tangential acceleration, which is equal to r α , where α is the angular acceleration of the body. v r ~ =
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