Machinery's Handbook, 31st Edition
192 Impulse and Momentum Example: A 1000-pound block is pulled up a 2-degree incline by a cable exerting a con stant force F of 600 lbs. If the coefficient of friction μ between the block and the plane is 0.5, how fast will the block be moving up the plane 10 seconds after the pull is applied? Solution: The resultant force R causing the body to be accelerated up the plane is the difference between F , the force acting up the plane, and P , the force acting to resist motion up the plane. This latter force for a body on a plane is given by the formula at the top of page 176 as P = W ( m cos a + sin a ) where a is the angle of the incline. Thus, R = F − P = F − W ( μ cos α + sin α ) = 600 − 1000(0.5 cos2 ° + sin 2 ° ) = 600 − 1000(0.5 3 0.99939 + 0.03490) R = 600 − 535 = 65 pounds. Formula (4c) can now be applied to determine the speed at which the body will be mov ing up the plane after 10 seconds: Rt W g -- V f W g -- V o – = 65 10 × 1000 32.2 ------ V f 1000 32.2 ------ 0 × – = V f 65 10 32.2 × × 1000 = ------------------ 20.9 ft/sec 14.3 mph = = A similar example using metric SI units is as follows: A 500 kg block is pulled up a 2 degree incline by a constant force F of 4 kN. The coefficient of friction μ between the block and the plane is 0.5. How fast will the block be moving 10 seconds after the pull is applied? The resultant force R is: cos sin R F Mg 4000 500 981 05 0 99939 003490 1378N or 1.378 kN # # µ α α = − + = − + = ^ ^ h h . . . . Formula (4c) can now be applied to determine the speed at which the body will be moving up the plane after 10 seconds. Replacing W / g by M in the formula, the calcu lation is:
Rt MV MV V V 1378 10 500 0 500 f # # = − = − = ^
1378 10 27 6 m/s f o f h
.
=
Angular Impulse and Momentum: In a manner similar to that for linear impulse and mo- ment, the formulas for angular impulse and momentum for a body rotating about a fixed axis are: (5a) (5b) where J M is the moment of inertia of the body about the axis of rotation in lb-ft-sec 2 , ω is the angular velocity in rad/sec, T o is the torque in lb-ft about the axis of rotation, and t is the time in seconds that T o acts. The change in angular momentum of a body is numerically equal to the angular impulse that causes the change in angular momentum: (5c) J Angular momentum M = ω T t Angular impulse o = T t J J J Angular Impulse Change in Angular Momentum o M f M o M f o ~ ~ ~ ~ = = − = − ^ h
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