Energy Practice Problems

Energy is a very abstract concept; therefore, it is extremely difficult to condense all information into one simple definition. Even world-class scientists have stated that there is no exact definition for what we all know as energy. Energy, best defined, is a substance-like quantity, transferred between parts of a/the system with the resulted production of work. Energy is a scalar. Think of energy like money, it is always there circulating around and it doesn't just grow on trees.

Energy Model Basics

• Is conserved when there is no work, radiation, heat, etc... 
• Cannot be created nor destroyed (this is another way of saying it's conserved). 
• Can move in our out of a system via work, heat, or radiation.  
• Is able to cause a change. 
• Can exist in many forms (E_g, E_k, E_t, E_el). 
• Can transform from one type within a system to another type within the system.  
• Has units: Joules (J) = kg m²/s², = Nm

Defining a System

A system is a part of the universe that is being considered. The types of energy which can be stored in a system depend on what objects are included in it. For example if your system doesn't have a spring (or other spring-like objects) then your system can't store elastic potential energy! Objects which might be included in the system are the object, Earth, ground, air, ramp, etc. Once a system boundary has been defined, then we can determine what types of energy can be stored in the system and we can also determine if energy is leaving/entering the system via working. If there are no external factors moving energy in/out of a system then energy must be conserved. 

Tip: Gravitational Potential Energy (E_g) is stored between separated objects which can fall towards each other. For a system to store significant amounts of E_g there needs to be a very large object like a planet, sun, moon etc. in your system. 

How to choose your system: 

Energy Storage

There are many ways to store energy. Energy is always stored in objects and each type of energy has a clear "red flag" indicator that the type of energy is stored in your system or not. The chart below summarizes some common ways energy is stored that we will study this year. 

Law of Conservation of Energy (1^st Law of Thermodynamics)

ΔE_{total stored} = W+ Q+ R

In words this means: to change the total energy in a system, energy must come into the system or else leave via one of three mechanisms: working, heating, or radiating. If no energy enters/leaves the system, then the total energy must remain constant since energy is not created or destroyed. Another way to think about this is if we choose the whole universe as our system, then by definition there cannot be any external factors transferring energy in/out, so the total energy in the universe is always constant. 

What are the three ways to move energy in/out of a system?

• Working (W): a mechanical transfer of energy which happens when external forces act parallel to displacements. 
• Heating (Q): a thermal transfer of energy. Example: a Bunsen burner rises the temperature of a beaker of water. Heat and temperature will be studied in Chemistry next year, but for now Q=0 J (i.e. we won't study heat). 
• Radiating (R): an electromagnetic transfer of energy. Examples: a microwave heats up soup or the sun shines on a solar panel. We are not studying radiation so R=0 J. 

In many cases there won't be any working, heating, or radiation. In these cases, energy is conserved so:

     ΣE_start = ΣE_{everyothertime}

Work

In physics this year we won't study heating (Q) or radiation (R) so those values will always be zero. Therefore, we can simplify the 1st Law of Thermodynamics to: 

This means that for our purposes, the only way to change the total energy of the system is to move energy in or out via work. Work happens when an external unbalanced force acts parallel (or partially parallel) to the displacement of an object in the system. It is very important to define your system before attempting to determine if work is done. Only a force acting on an object in your system by an object outside your system can transfer energy in or out of the system. 

Although work is a scalar, work can either be positive or negative. When the external force is parallel to the object's displacement the work is positive and the total energy of the system increases. When the external force is in the opposite direction as the object's displacement the work is negative and the total energy of the system decreases. If the external force is perpendicular to the object's displacement the work is zero and the total energy of the system doesn't change. 

Net work is the sum of work. For example, it could be that there is work from an outside force that takes 10 J from the system (force is opposite displacement) and another external force that gives 10 J to the system (force is parallel to displacement) so the net work would be zero and the total energy of the system would remain constant. 

Tip: if you are able to define your system so that there are no external net forces (i.e. no net work) this usually makes solving the question easier. However, in certain cases it's easier to leave an object outside of the system, even though it does work on the system, because the object is very complicated to analyze (like a person). 

Example 1:  

In the system to the left, consisting of the ball, the earth, and the ramp, the ball's displacement (up the ramp) is up and left. Let's assume the ball is being pushed at a constant velocity. Since the hand is not in the system, it may do work because it is an outside force. The force by the hand is in the same direction as the ball's displacement, thus making the work positive. This means that the energy of the ball, earth, and ramp system increases. This makes sense since the kinetic energy of the system would be constant while the gravitational potential energy of the system rises. No type of energy in this system decreases over time.

Example 2: 

Let's consider a situation where the external force and displacement are perpendicular to each other. Define a system of just the cake as shown below. The cake is moved to the right at a constant velocity. A waiter hold the cake up on a tray. 

Even though there is an external force on the cake system (Fn on cake by tray), no work done and the energy system does not change. The reason for this is that the normal force is completely perpendicular to the displacement of the cake, so it doesn't transfer any energy to the cake system. This makes sense since the cake system only stores Ek at the beginning (no Eg since Earth is not in the system). Since the cake is moving at a constant velocity, Ek stays the same and no other types of energies change. Thus the total energy of the system is constant.

Quantitative Energy 

Spring Lab

In the spring lab we stretched a spring to increasing displacements while measuring the spring force (i.e. how much force it took to pull). We used the spring force and spring displacements to make connections with work and energy. 

Energy Pie Charts and Bar Graphs

Energy pie charts show how the types of energy stored in a system change over time. If there is no work, the size of the energy pie will remain constant. If there is work being done, it's typically wise to draw bar graphs instead of pie charts. If both the total amount of energy and some of the types of energies are changing pie charts tend to be confusing. 

Example 1: 

Watch this video of a duck climbing up a ladder. Then consider a system of the duck and Earth and draw at least 3 pie charts that show how the energy of the duck/Earth system change over time. 

Explanation The toy is winded up therefore the duck is deformed. This means that the duck starts with elastic energy. Over time the duck unwinds so the elastic potential energy drops. It's also clear from the video that the duck is moving at constant speed (KINETIC ENERGY), therefore that part of the pie chart will stay the same. Finally, as the duck is slowly unwinding, it is moving UPWARDS (GRAVITATIONAL ENERGY increases). Since there are no external forces parallel to the duck's displacement we know that the total energy remains constant.

Example 2:

Consider a toy popper which starts out deformed on the desk then shoots upwards before finally landing back on the desk. For a system of the popper, table, Earth, draw energy bar graphs before launch, at the instant after launch, at the highest point, half way down, and when the popper is resting on the table again. 

You could also represent this problem with charts like this:

Power

Power is a measure of how quickly energy is moved in or out of a system. The unit of power is Watts or J/s.

2^nd Law of Thermodynamics

Energy Efficiency

Energy Efficiency is the ratio of useful energy to the total energy.

Energy Music Video (General Review)