Machinery's Handbook, 31st Edition
Force
187
P
P
W cos
W cos
W
Fig. 1a. Fig. 1b. In preparing a free body diagram, it is important to understand that only those forces exerted on the body being considered are shown; forces exerted by the body on other bodies are disregarded. This feature makes the free body diagram an invaluable aid in the solution of problems in mechanics. Example: A 100-pound body is being hoisted by a winch, the tension in the hoisting cable being kept constant at 110 pounds. At what rate is the body accelerated? Solution: Two forces are acting on the body, its weight, 100 pounds downward, and the pull of the cable, 110 pounds upward. The resultant force R , from a free body diagram, is therefore 110 − 100. Thus, applying Newton’s second law,
. a 110 100 32 16 100 100 a # − =
110 lb
W
. 32 16 10 3 216 ft
upward 2
.
/sec
100 lb = It should be noted that since in this problem the resultant force R was positive (110 − 100 = +10), the acceleration a is also positive, that is, a is in the same direction as R , which is in accord with Newton’s second law. = Example using SI metric units: A body of mass 50 kilograms is being hoisted by a winch, and the tension in the cable is 600 newtons. What is the acceleration? The weight of the 50 kg body is 50 g newtons, where g = approximately 9.81 m/s 2 (see Note on page 194). Applying the formula R = Ma , the calculation is: (600 − 50 g ) = 50 a . Thus, . . a g 50 600 50 50 600 50 9 81 2 19 m/s 2 # = − = − = ^ h Formulas Relating Torque and Angular Acceleration: For a body rotating about a fixed axis the relation between the unbalanced torque acting to produce rotation and the result- ing angular acceleration may be determined from any one of the following formulas, each based on Newton’s second law:
T J T Mk T o M o o = = =
α
o 2
α
Wk
. Wk 3216 α o 2
g o 2
α
=
where T o is the unbalanced torque in pounds-feet; J M in ft-lbs-sec 2 is the moment of inertia of the body about the axis of rotation; k o in feet is the radius of gyration of the body with respect to the axis of rotation, and α rad/sec 2 , is the angular acceleration of the body. Example: A flywheel has a diameter of 3 feet and weighs 1000 pounds. What torque must be applied, neglecting bearing friction, to accelerate the flywheel at the rate of 100 revolu tions per minute per second (rpm/s)?
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