Basic Thermodynamics ~ J. Pohl © | www.THERMOspokenhere.com (116-C255) |
Basic mechanics can be used with the energy equation to evaluate the efficacies of tasks. Consider the manners by which each of two workers moves a BOX of mass 12 kilograms from the storage floor to the tailgate of a truck. Use the data of the figures.
Calculate i) the Power (Watts), ii) the Energy Expended (J), for the moving events of A and B as described below.
Worker A:
From the floor (1), the BOX is lifted waist-high, then carried and placed on truck tailgate (2).
♦ We select a BOX as our system. We model it as an EXTENDED BODY. We write the rate form of the energy equation:
![]() | (1) Anticipate zero change of internal energy. |
Next separate variables and apply the integration operator ( ∫). The limits are the time of initiation (call that time "0+") and a second time, an increment of time later, t = t + Δt = 0 + Δt.
![]() | (2) Integration operator and limits are applied. |
The energies, left-of-equality, integrate directly since their integrand are unity (one or "1"). Mathematics calls differentials of integrals having "1" as the integrand, Exact differentials.
The work, right-of-equality, will require some thought. The summation (Σ) reminds us to identify all applicable manners of work of the event. There are two forces that act on the BOX. The first and larger is the force applied by the person who moves it. A second force is the force of drag, the force of resistance exerted by the surrounding air as the BOX moves through it. The work against "drag forces" is sufficiently small to be ignored (and difficult to calculate when were we to include it). Notice since we use the perspective, potential energy, our system is in fact the box and Earth. Consequently the gravity force is not considered - it's effect is potential energy change. With these considerations our equation becomes (3):
![]() | (3) Left of equality integrates to become increments of energy. |
In Equations (1), (2) and (3), we expressed work not as W-dot but a W(t)-dot (Must use "-dot" because HTML does not have over-dot). This makes the point that of the work of the mover over the duration of the event (Δt) might be more in the early seconds and less later. In general work is a function of time. Given our data, we cannot perform the integration. To proceed (to get any answer at all) we must apply the Mean Value Theorem of Calculus to the work integral.
![]() | (4) ΔKE could be large. |
Of the three terms, term (3) is determined last (as the sum of terms (1) and (2)). Term (2) evaluates readily. It is simply the mass times gravity times the height of lift.
![]() | (5) The kinetic energy could be very large. |
What about kinetic energy; can that term be calculated or approximated? To write the kinetic energy term, we require the speed of the BOX. From the sketch we have the very least distance the BOX must travel and the time required. The equation relating these is:
![]() | (6) We can calculate the least average speed. |
The least distance the BOX is moved in 12 seconds is [(3.6m)2 + (1.5m)2]1/2 = 3.9 m. Thus the kinetic energy the BOX must attain (at its very least) is written in (7) as:
![]() | (7) Least kinetic energy. |
When numbers are applied to (7) the least average power expended by Worker A is obtained:
![]() | (8) Least average power of Worker A. |
#####################################
Next we investigate the task as done by Worker B. The image (below right)shows characteristics of the action as he moves a BOX onto the truck tailgate.
Worker B:
From the floor (1) needing 2 seconds the BOX is lifted waist-high (2). Carrying toward the truck requires 4 seconds but the BOX is dropped (3). Lifted again to waist-height, in 4 seconds, the BOX is placed on the tailgate.
The BOX location and properties, and event duration are the same as for Worker A. Results of that analysis can be used (with some thought).
We can use information of the second line of (8). Worker B lifts the BOX twice and he squanders its kinetic energy when he drops it. For Worker B we have:
![]() | (9)
It is helpful NOT to rearrange the terms until the last step. |
Basic mechanics can be used with the energy equation to evaluate the efficacies of tasks. Consider the manners by which each of two workers moves a BOX of mass 12 kilograms from the storage floor to the tailgate of a truck. Use the data of the figures.
Calculate i) the Power (Watts), ii) the Energy Expended (J), for the moving events of A and B as described below.