Velocity and Acceleration Machinery's Handbook, 31st Edition
183
Table 1. Linear Motion with Constant Acceleration
To Find Known
Formula
To Find Known
Formula
Motion Uniformly Accelerated from Rest (V o = 0)
a , t
S , V f
t = 2 S ÷ V f
S = 1 ∕ S = 1 ∕
2
2 at
S , a
V f , t V f , a
S
t
2 V f t
S a 2 ' =
t
a , V f
2 ÷ 2 a
t = V
S = V f V f = at
f ÷ a
a , t S , t a , S
S , t
a = 2 S ÷ t 2
S , V V f , t
a = V f
V f = 2 S ÷ t V aS 2 f =
2 ÷ 2 S
a
a = V f ÷ t
V f
Motion Uniformly Accelerated from Initial Velocity V o
a , t , V o V o , V f , t V f , a , t V o , a , t V o , S , t V o , a , S
V o , V f , a t = ( V f − V o ) ÷ a V o , V f , S t = 2 S ÷ ( V f + V o )
S = V o t +
1 ∕
2
2 at
t
S = ( V f + V o ) t ÷ 2
S
V o , V f , a S = ( V f
V o , V f , S a = ( V f
2 ) ÷ 2 a
2 ) ÷ 2 S
2 − V
2 − V
o
o
V o , V f , t V o , S , t V f , S , t
f − V o ) ÷ t
S = V f t −
a = ( V
1 ∕
2
2 at
a
V f = V o + at
a = 2( S − V o t ) ÷ t 2 a = 2( V f t − S ) ÷ t 2
V f = (2 S ÷ t ) − V o V V aS 2 f o 2 = +
Meanings of Symbols
S , a , t
V f = ( S ÷ t ) +
1 ∕
2 at
V f
S = distance moved in ft or m V f = final velocity, ft/s or m/s V o = initial velocity, ft/s or m/s a = acceleration, ft/s 2 or m/s 2 t = time of acceleration, sec or s
V f , a , S V f , S , t V f , a , t
V V aS 2 o f 2 = − V o = (2 S ÷ t ) − V f
V o = V f − at V o = ( S ÷ t ) −
S , a , t
V o 2 at Since V o , V f , and t are known, a can be determined from the formula . . . /s a V V t 1389 2778 2 695m f o 2 ' ' = − = − = − ^ ^ h h The time required to stop the car can be determined from the formula . . t V V a 0 2778 6 95 4 seconds f o ' ' = − = − − = ^ ^ ^ h h h The distance traveled by the car is obtained from the formula 1 ∕
1 2 -- at
1 2 -- –6.95 ( ) 4 2
2
S V o t = = Angular Velocity of a Rotating Body.— The angular velocity of a rotating body is the angle through which the body turns in a unit of time. Angular velocity is commonly ex- pressed in terms of revolutions per minute rpm, but in certain engineering applications it is necessary to express it as radians per second (rad/s). By definition there are 2 π radians in 360 degrees, or one revolution, so that one radian = 360 / 2 π ≈ 57.3 degrees. To convert angular velocity n in rpm to angular velocity ω in radians per second, and vice versa, use Equation (1): (1) ω π n 30 = --- rad/s n 30 ω π = ----- rpm + 27.78 ( ) 4 ( ) + 111.12 – 55.6 ( ) 55.52 m = =
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