Machinery's Handbook, 31st Edition
FORCE OF A BLOW 191 pile is driven. Since the impact force is not accurately known, an average value, called the “average force of the blow,” may be assumed. Equating the work done on the pile and the work done by the falling body, which in this case is a pile driver: d WS Average force of blow # = d WS Average force of blow = where S = total height in feet through which the driver falls, including the distance d that the pile is driven W = weight of driver in pounds d = distance in feet which pile is driven or, When using metric SI units, it should be noted that a body of mass M kilograms has a weight of Mg newtons, where g = approximately 9.81 m/s 2 . If the body falls a distance S meters, it can do work equal to MgS joules. The average force of the blow is MgS / d newtons, where d is the distance in meters that the pile is driven. Example: A pile driver weighing 200 pounds strikes the top of the pile after having fallen from a height of 20 feet. It forces the pile into the ground a distance of 1 ∕ 2 foot. Before the ram is brought to rest, it will do 200 3 (20 + 1 ∕ 2 ) = 4100 foot-pounds of work, and as this energy is expended in a distance of 1 ∕ 2 foot, the average force of the blow equals 4100 ÷ 1 ∕ 2 = 8200 pounds. A similar example using metric SI units is as follows: A pile driver of mass 100 kilo grams falls 10 meters and moves the pile a distance of 0.3 meter. The work done = 100 g (10 + 0.3) joules, and it is expended in 0.3 meter. Thus, the average force is . . . g kN 03 100 10 3 33680 3368 newtons or # = Impulse and Momentum.— The linear momentum of a body is defined as the product of the mass M of the body and the velocity V of the center of gravity of the body:
Linear momentum MV or since M W g = -- = Linear momentum WV g = -----
(4a)
It should be noted that linear momentum is a vector quantity, the momentum being in the same direction as V . Linear impulse is defined as the product of the resultant R of all the forces acting on a body and the time t that the resultant acts: (4b) The change in the linear momentum of a body is numerically equal to the linear impulse that causes the change in momentum: (4c) where V f , the final velocity of the body after time t , and V o , the initial velocity of the body, are both in the same direction as the applied force R . If V o and V f are in opposite directions, then the minus sign in the formula becomes a plus sign. In metric SI units, the formulas are: Linear Momentum = MV kg·m/s, where M = mass in kg, and V = velocity in meters per second; and Linear Impulse = Rt newton- seconds (N·s), where R = force in newtons, and t = time in seconds. In Formula (4c) above, W / g is replaced by M when SI units are used. Rt Linear Impulse = Rt g W V V Linear Impulse change in Linear Momentum f o f o g W V g W V = = − = − ^ h
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