69 Work to Lift A Patient

Energy Conservation in A System

In the previous chapters we analyzed the changes in kinetic energy of an entire system due to a net external work. That allowed us to calculate fall speeds, impact forces, stopping distances, and work required to move objects.  To analyze the internal functioning of the human body and other complex systems we will also want to keep track energy transfer energy between categories within an isolated system, when when the total energy does not change and the net external work is zero. Those internal categories include potential energy in the form of gravitational potential energy (PEg), elastic (spring) potential energy (PEs), and chemical potential energy (PEc). The internal kinetic energy of the system (KEi) is stored in the motion of the internal components of the system.  The microscopic kinetic energy stored in the random motion of the individual microscopic particles that make up the system is often separated out and called thermal energy (TE).

The Principle of Energy Conservation states that energy (E) cannot be created or destroyed. That means internal forces between components within a system can do internal work that transfers energy between the various categories internal to a system, but the total internal energy of an isolated system cannot change. In that case, adding up the all of the internal energy changes we should get a total change of zero, which written as an equation is known as the Law of Conservation of Energy

(1)   \begin{equation*} 0 = \Delta KE_i + \Delta TE +\Delta PE_g + \Delta PE_s + \Delta PE_c \end{equation*}

The equation above looks long, but in many situations we can neglect most of the terms and the equation will become much shorter.  In order to determine which energies in our system are changing and which are not, we need to learn a little more about each one. Let’s start with gravitational potential energy.

Gravitaional Potential Energy

If we want to analyze the gravitational potential energy in a system then we cannot treat the gravitational force between the objects in the system as external to the system. Instead, the work done by the gravitational force will change the system’s gravitational potential energy. Let’s look at an example to see how that works out.

Everyday Example: Lifting a Patient

Jolene works with two other nurses to smoothly lift a patient that weighs 75 kg a distance of 0.20 m at constant speed.  

  • How much work was done on the patient by the nurses?
  • How much did the gravitational potential energy change?

We are going to want to know the upward force required to lift the patient at constant speed. We will first choose our system to be the patient alone.

(2)   \begin{equation*} W_{on} = \Delta KE \end{equation*}

Moving at constant speed, the kinetic energy of the patient does not change so we know the net work must be zero. The nurses do positive work that would increase the kinetic energy of the patient, but acting opposite to the motion, gravity does an equal amount of negative work to transfer that energy into gravitational potential energy instead. The change in gravitational potential energy is equal to the negative of work done by gravity:

(3)   \begin{equation*} \Delta PE_g = -W_g \end{equation*}

The work done by gravity to move energy into gravitational potential energy instead of kinetic energy can be calculated using the work equation W = Fdcos\theta

(4)   \begin{equation*} \Delta PE_g = - Fdcos\theta \end{equation*}

The force of gravity can be calculated as F_g = mg, the distance is the height increase of the patient \Delta h, and the angle is 180^{\circ} because gravity acts opposite to motion as the patient rises.

(5)   \begin{equation*} \Delta PE_g = - mg\Delta h cos(180^{\circ}) \end{equation*}

We know cos(180^{\circ}) =-1, so the negatives cancel:

(6)   \begin{equation*} \Delta PE_g = mg\Delta h \end{equation*}

Near Earth’s surface we can always calculate the change in potential energy using the formula above. This formula will automatically gives a decrease in gravitational potential energy when an object is lowered or falls because the change in height will be negative. Entering our specific values for this example:

(7)   \begin{equation*} \Delta PE_g (75\, \bold{kg} )(9.8\,\bold{m/s/s})(0.20\,\bold{m})= 147\,\bold{J}) \end{equation*}

The gravitational potential energy increased by about 147 J. We know the nurses do an equal amount of positive work to transfer at least that much energy out of their bodies. The body is only about 20% efficient so the nurses actually transferred 5x more chemical potential energy out of her body, 80% of that as thermal energy, and all of which originally entered her body as chemical potential energy in food.

Reinforcement Exercises

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