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Figure 8.6. A schematic diagram of a box containing a computer packaged for shipment. If the box is dropped from a height h, the mechanical behavior of the computer will be similar to that of a mass m falling on a spring with downward velocity v.

The maximum spring force acting on the computer is then equal to

The peak force during the collision increases with the falling height, the mass of the computer, and the stiffness of the spring used as cushioning. Thus, the more compliant the spring, the smaller is the peak force. However, if the spring is chosen to be very compliant, the computer might actually hit the box sitting on the floor, and then the equation for the peak force just given will not be correct.

Example 8.3. Pendulum Hit by a Bullet. A ballistic pendulum is used to determine the speed of rifle bullets. This pendulum is a rectangular block of mass m2 that is supported by two cords (Fig. 8.7). A bullet of mass m1 and velocity v1 strikes the pendulum at time t1 and becomes embedded in the block. Develop an equation that relates the velocity of the bullet to the amplitude of the pendulum swing. Figure 8.7. Schematic diagram of a rectangular block of mass m2 that is supported on two cables. A bullet of mass m1 and velocity v1 strikes the mass m2 at an arm length L.

Solution: The linear momentum before the impact must be equal to the linear momentum after the impact and thus:

The kinetic energy of the bullet-pendulum-inextensible cord complex right after the impact is

The inextensible cord produces no work on the mass m2 because the tensile force exerted by the cord is perpendicular to the the velocity of the point to which it is attached. Conservation of mechanical energy of the system causes the kinetic energy right after the impact to transform into potential energy at the end of the pendulum swing:

T2 + V2 = T3 + V3 = (m1 + m2) gL (1 - cos 0) (8.24b)

in which V2 = T3 = 0. Solving Eqns. 8.24a and 8.24b for v12, we obtain the following equation that relates the velocity of the bullet before it strikes the pendulum to the maximum swing of the pendulum which occurs as a result of the impact:

Knowing the values of L, m1, and m2, one can determine the incoming speed of the bullet by measuring the maximum amplitude of the pendulum. 