MCAT Physics and Math Review

Chapter 2: Work and Energy

Introduction

The Greek myth of Sisyphus is a tale of unending, pointless work. For eternity, Sisyphus was sentenced to roll a large, heavy rock up a steep hill as penance for his crimes. Just as Sisyphus would nearly reach the top of the hill, the rock would roll back again to the bottom. The cycle continued for eternity: Sisyphus would near the top of the hill, and the boulder—enchanted by Zeus—would roll away, forcing Sisyphus to restart his task.

This is a story of work and mechanical energy transfer. Pushing that boulder up the hill, Sisyphus exerted forces that performed work on the rock, resulting in an increase in the rock’s gravitational potential energy. When the rock escaped from his grasp and rolled backwards, its energy changed from gravitational potential energy into kinetic energy. While Sisyphus’s punishment was futile work, it serves as a strong model of the exchange of mechanical energy between its two forms: potential and kinetic. While a number of other forms of energy exist (thermal energy, sound, light, chemical potential energy, and electrical potential energy, to name a few), mechanical energy specifically focuses on objects in motion.

This chapter reviews the fundamental concepts of energy and work. The work–energy theorem is a powerful expression of the relationship between energy and work which is often a simpler approach to kinematics questions on Test Day. Finally, we’ll discuss the topic of mechanical advantage, and we’ll examine how a pulley or ramp might be helpful in raising heavy objects. We hope to convince you throughout the Kaplan MCAT program that your preparation for Test Day is in no way a Sisyphean task.

2.1 Energy

Energy refers to a system’s ability to do work or—more broadly—to make something happen. This broad definition helps us understand that different forms of energy have the capacity to perform different actions. For example, mechanical energy can cause objects to move or accelerate. An ice cube sitting on the kitchen counter at room temperature will absorb thermal energy through heat transfer and eventually melt into water, undergoing a phase transformation from solid to liquid. Let’s turn our attention to the different forms that energy can take. After that, we will discuss the two ways in which energy can be transferred from one system to another.

KINETIC ENERGY

Kinetic energy is the energy of motion. Objects that have mass and that are moving with some speed will have an associated amount of kinetic energy, calculated as follows:

Equation 2.1

where K is kinetic energy, m is the mass in kilograms, and ν is speed in meters per second. The SI unit for kinetic energy, as with all forms of energy, is the joule (J), which is equal to 

KEY CONCEPT

Kinetic energy is incredibly important on the MCAT; any time an object has a speed, think about kinetic energy and link its kinetic energy to the related concepts of work and conservation of mechanical energy.

Recall the falling objects in Chapter 1. Such objects have kinetic energy while they fall. The faster they fall, the more kinetic energy they have. Be mindful of the fact that the MCAT is interested in testing students’ comprehension of the relationship between kinetic energy and speed. From the equation, we can see that the kinetic energy is a function of the square of the speed. If the speed doubles, the kinetic energy will quadruple, assuming the mass is constant. Also note that kinetic energy is related to speed—not velocity. An object has the same kinetic energy regardless of the direction of its velocity vector.

KEY CONCEPT

Kinetic energy is related to speed, not velocity. An object has the same kinetic energy regardless of the direction of its velocity vector.

Falling objects have kinetic energy, but so do objects that are moving in other ways. For example, the kinetic energy of a fluid flowing at some speed can be measured indirectly as the dynamic pressure, which is one of the terms in Bernoulli’s equation—discussed in Chapter 4 of MCAT Physics and Math Review. Objects that slide down inclined planes gain kinetic energy as their speeds increase down the ramp.

Example:

A 15 kg block, initially at rest, slides down a frictionless incline and comes to the bottom with a speed of  as shown below. What is the kinetic energy of the object at the top and bottom of the ramp?

Solution:

At the top, v = 0, so the kinetic energy is

At the bottom, the kinetic energy is

POTENTIAL ENERGY

Potential energy refers to energy that is associated with a given object’s position in space or other intrinsic qualities of the system. Potential energy is often said to have the potential to do work, and can take named forms. Energy can be stored as chemical potential energy—this is the energy we absorb from the food we eat when we digest and metabolize it. Electrical potential energy, which is discussed in Chapter 5 of MCAT Physics and Math Review, is based on the electrostatic attractions between charged particles. In this chapter, we’ll examine the types of potential energy that are dissipated as movement: gravitational potential energy and elastic potential energy.

Gravitational Potential Energy

Gravitational potential energy depends on an object’s position with respect to some level identified as the datum (“ground” or the zero potential energy position). This zero potential energy position is usually chosen for convenience. For example, you may find it convenient to consider the potential energy of the pencil in your hand with respect to the floor if you are holding the pencil above the floor or with respect to a desktop if you are holding the pencil over a desk. The equation that we use to calculate gravitational potential energy is

U = mgh

Equation 2.2

where U is the potential energy, m is the mass in kilograms, g is the acceleration due to gravity, and h is the height of the object above the datum.

MCAT EXPERTISE

The height used in the potential energy equation is relative to whatever the problem states is the ground level. It will often be simply the distance to the ground, but it doesn’t need to be. The zero potential energy position may be a ledge, a desktop, or a platform. Just pay attention to the question stem and use the height that is discussed.

Potential energy has a direct relationship with all three of the variables, so changing any one of them by some given factor will result in a change in the potential energy by the same factor. Tripling the height—or tripling the mass of the object—will increase the gravitational potential energy by a factor of three.

Example:

An 80 kg diver leaps from a 10 m cliff into the sea, as shown below. Find the diver’s potential energy at the top of the cliff and when he is two meters underwater, using sea level as the datum.

Solution:

At the top of the cliff:

When he is two meters underwater:

Elastic Potential Energy

Springs and other elastic systems act to store energy. Every spring has a characteristic length at which it is considered relaxed, or in equilibrium. When a spring is stretched or compressed from its equilibrium length, the spring has spring potential energy, which can be determined by

Equation 2.3

Where U is the potential energy, k is the spring constant (a measure of the stiffness of the spring), and x is the magnitude of displacement from equilibrium. Note the similarities between this equation and the formula for kinetic energy.

TOTAL MECHANICAL ENERGY

The sum of an object’s potential and kinetic energies is its total mechanical energy. The equation is

E = U + K

Equation 2.4

where E is total mechanical energy, U is potential energy, and K is kinetic energy. The first law of thermodynamics accounts for the conservation of mechanical energy, which posits that energy is never created nor destroyed—it is merely transferred from one form to another. This does not mean that the total mechanical energy will necessarily remain constant, though. You’ll notice that the total mechanical energy equation accounts for potential and kinetic energies but not for other forms of energy, such as thermal energy that is transferred as a result of friction (heat). If frictional forces are present, some of the mechanical energy will be transformed into thermal energy and will be “lost”—or, more accurately, dissipated from the system and not accounted for by the equation. Note that there is no violation of the first law of thermodynamics, as a full accounting of all the forms of energy (kinetic, potential, thermal, sound, light, and so on) would reveal no net gain or loss of total energy, but merely the transformation of some energy from one form to another.

CONSERVATION OF MECHANICAL ENERGY

In the absence of nonconservative forces, such as frictional forces, the sum of the kinetic and potential energies will be constant. Conservative forces are those that are path independent and that do not dissipate energy. Conservative forces also have potential energies associated with them. On the MCAT, the two most commonly encountered conservative forces are gravitational and electrostatic. Elastic forces can also be approximated to be conservative in many cases, although the MCAT may include spring problems in which frictional forces are not ignored (in actuality, springs heat up as they move back and forth due to the friction between the particles of the spring material). There are two equivalent ways to determine whether a force is conservative, as demonstrated in Figure 2.1.

Figure 2.1. Determining if a Force is Conservative If the change in energy around any round-trip path is zero—or if the change in energy is equal despite taking any path between two points—then the force is conservative.

BRIDGE

The transfer of energy from one form to another is a key feature of bioenergetics and metabolism, discussed in Chapters 9 through 12 of MCAT Biochemistry Review. When looking at carbohydrate metabolism, one can see the chemical potential energy in the bonds in glucose being converted into electrical potential energy in the high-energy electrons of NADH and FADH2, which is dissipated along the electron transport chain to generate the proton-motive force (another example of electrical potential energy). This force fuels ATP synthase, trapping the energy in high-energy phosphate bonds in ATP.

One method is to consider the change in energy of a system in which the system is brought back to its original setup. In mechanical terms, this means that an object comes back to its starting position. If the net change in energy is zero regardless of the path taken to get back to the initial position, then the forces acting on the object are conservative. Basically, this means that a system that is experiencing only conservative forces will be “given back” an amount of usable energy equal to the amount that had been “taken away” from it in the course of a closed path. For example, an object that falls through a certain displacement in a vacuum will lose some measurable amount of potential energy but will gain exactly that same amount of potential energy when it is lifted back to its original height, regardless of whether the return pathway is the same as that of the initial descent. Furthermore, at all points during the fall through the vacuum, there will be a perfect conversion of potential energy into kinetic energy, with no energy lost to nonconservative forces such as air resistance. Of course, in real life, nonconservative forces are impossible to avoid.

The other method is to consider the change in energy of a system moving from one setup to another. In mechanical terms, this means an object undergoes a particular displacement. If the energy change is equal regardless of the path taken, then the forces acting on the object are again all conservative.

When the work done by nonconservative forces is zero, or when there are no nonconservative forces acting on the system, the total mechanical energy of the system (U + K) remains constant. The conservation of mechanical energy can be expressed as

ΔE = ΔU + ΔK = 0

Equation 2.5

where ΔE, ΔU, and ΔK are the changes in total mechanical energy, potential energy, and kinetic energy, respectively.

KEY CONCEPT

Conservative forces (such as gravity and electrostatic forces) conserve mechanical energy. Nonconservative forces (such as friction and air resistance) dissipate mechanical energy as thermal or chemical energy.

When nonconservative forces, such as friction, air resistance, or viscous drag (a resistance force created by fluid viscosity) are present, total mechanical energy is not conserved. The equation is

Wnonconservative = ΔE = ΔU + ΔK

Equation 2.6

where Wnonconservative is the work done by the nonconservative forces only. The work done by the nonconservative forces will be exactly equal to the amount of energy “lost” from the system. In reality, this energy is simply transformed into another form of energy, such as thermal energy, that is not accounted for in the mechanical energy equation. Nonconservative forces, unlike conservative forces, are path dependent. The longer the distance traveled, the larger the amount of energy dissipated.

Example:

A baseball of mass 0.25 kg is thrown in the air with an initial speed of  but because of air resistance, the ball returns to the ground with a speed of  Find the work done by air resistance.

Solution:

Air resistance is a nonconservative force. To solve this problem, the energy equation for a nonconservative system is needed. The work done by air resistance is:

Wnonconservative = ΔE = ΔU + ΔK

In this case, ΔU = 0 because the initial and final heights are the same. Therefore,

The negative sign in the answer indicates that energy is being dissipated from the system.

MCAT Concept Check 2.1:

Before you move on, assess your understanding of the material with these questions.

1.    Define kinetic energy and potential energy.

·        Kinetic energy:

·        Potential energy:

2.    Compare and contrast conservative and nonconservative forces:

 

Conservative Forces

Nonconservative Forces

What happens to total mechanical energy of the system?

   

Does the path taken matter?

   

What are some examples?