## Conservation of Mechanical Energy

As a rigid body moves from a position at time t1 to another position at time t2, the change in its kinetic energy is given by the following relation:

in which T2 and T1 are the kinetic energy of the body at times t2 and t1, and W1-2 is the work done during the time between t1 and t2. Part of the work done may be the result of conservative forces acting on the rigid body:

in which W'1-2 represents the work done by forces other than gravity and springs.

If all the external forces that do work on the rigid body are conservative, then the conservation of mechanical energy holds:

Thus, the sum of kinetic and potential energy of a rigid body is constant unless work is done on the body by dissipative forces.

Example 8.2. Delivery Person Drops a Box Containing a Laptop from a Height h. The dynamics of a fall of a box containing a computer may be modeled as a mass of m striking a spring of stiffness k with velocity v (Fig. 8.6). When a box containing a computer is dropped from a height h, the mass-spring system is subjected to an impact. Develop an equation for the peak spring force produced when the delivery person drops the box from height h.

Solution: We assume that the box comes to rest as soon as it strikes the floor. The mass m of the computer in the box exerts zero force on the spring at the instant of impact. Nevertheless, the computer has a downward velocity v. This velocity can be found by considering the conservation of mechanical energy of mass m:

When the mass-spring system comes to rest at its lowest position, the decrease of kinetic energy must equal to the increase in the potential energy of the spring: 