Key Terminology
Energy: the quantitative property that must be transferred to a body or physical system to perform work on the body, or to heat it.
Mechanical Energy: Potential energy (U) and Kinetic energy (K)
System: A collection of objects we are focusing on
Law of Conservation of Energy: Energy can be converted in form, but not created or destroyed. The total amount of energy in the universe stays the same.
Types of Energy:
Mechanical Energy: Potential energy (U) and Kinetic energy (K)
System: A collection of objects we are focusing on
Law of Conservation of Energy: Energy can be converted in form, but not created or destroyed. The total amount of energy in the universe stays the same.
Types of Energy:
- Gravitational Potential: Energy stored in the Earth's gravitational field. Measured by the height of an object from the Earth's surface, the higher it is, the more gravitational potential energy. THERE IS ONLY GRAVITATIONAL POTENTIAL ENERGY WHEN THE EARTH IS IN THE SYSTEM.
- Kinetic: The energy stored in moving objects, measured by speed.
- Thermal/Internal: The energy stored in random moving atoms in a system. Measured by temperature.
- Elastic Potential: Energy stored in an elastic material by stretching/compressing it. Measured by the distance from equilibrium.
Representing Energy Transfers with Bar (LOL) Charts
Energy transfers can be represented by LOL charts. The left L represents the amount of initial energies, the middle O represents the system, and the right L represents the amount of energies final.
Energy is always conserved, but can be transferred between systems in three ways: Working (W), Heating (Q), and Radiating (R). Thus we can write the following energy conservation equation: Einitial + W + Q + R = Efinal. We can visualize this transfer with the LOL chart.
In this example, an ice cube cools a cup of coffee. Notice our system here is only the coffee since we are only interested in the energy flow of the coffee. The internal or thermal energy of the coffee is transferred away through heating and the coffee has less thermal energy in the end. Notice the blocks of energy also represent energy conservation: 4 at the start, 2 transferred away by heating, leaving 2 in the end.
Energy Problem Solving
Since we know that energy is conserved or transferred, we can use energy conservation equations and LOL charts to solve problems whenever there is a transfer of energy.
But before that, we need to know some key formulas
Consider the following problem:
But before that, we need to know some key formulas
- Elastic Potential energy (Us) = 1/2 kx^2 (where x is the distance from equilibrium)
- Gravitational Potential Energy (Ug) = mgh
- Kinetic Energy (KE) = 1/2 mv^2
Consider the following problem:
For simplicity, we can define zero gravitational potential energy as the height of muzzle, thus the system has a negative gravitational potential energy at the start and a lot of elastic potential energy. Our system here is the spring, ball and Earth. Notice the Earth needs to be included here to be able to have gravitational potential energy. In the final position, the ball is motionless at its highest point, with all the energy as gravitational potential. This also gives us the conservation of energy equation Us + Ug = Ugfinal, or, 1/2 kx^2 + mgh initial = mgh final. Plugging in values for k, x and m, we know that 1/2 * 667 * 0.25^2 + 1.5 * 9.8 * -0.25 = 1.5 * 9.8 * h final. Notice here our h initial is -0.25m, that is how much lower the ball is at the start when the spring is compressed. Calculating, this gives us 17.165 = 1.5 * 9.8 h final, and h = 1.17m. The max height the ball reaches is 1.17 meters.
Here is an extension to this problem:
Here is an extension to this problem:
Again, the height of zero is defined as the height of the muzzle, so there is negative gravitational potential energy at the start. We can construct an energy conservation equation of Us + Ug = Ek, or, 1/2 kx^2 + mgh = 1/2 * m*v^2. Plugging in values, we get 17.165 = 0.75 v^2, and v = 4.78 m/s. The muzzle velocity of the ball is 4.78 meters per second.
Work
Energy can be transferred by working, or, an external push/pull on a system. We can identify work occurring whenever there is a force applied on a system. Work can be found by the equations W = F . x or W = F . t. Since work is a scalar quantity, we take the dot product and only care about the component of the force in the direction of displacement. Therefore, any forces perpendicular to the direction of displacement will not affect work.
For example, if you move a block horizontally in the air, you are actually not doing work. The force on the block is only the force of gravity, which is vertical whereas the displacement is horizontal. They are perpendicular so there is no work.
Positive Work
When a transfer of energy occurs through positive working, energy is transferred into the system and Efinal > Einitial.
Negative Work
When a transfer of energy occurs through negative working, energy is transferred out of the system and Efinal < Einitial. For example, when catching a dropped object, you exert a normal force (up) opposite of the direction of displacement (down), doing negative work since the block now has less gravitational potential energy.
Positive Work
When a transfer of energy occurs through positive working, energy is transferred into the system and Efinal > Einitial.
Negative Work
When a transfer of energy occurs through negative working, energy is transferred out of the system and Efinal < Einitial. For example, when catching a dropped object, you exert a normal force (up) opposite of the direction of displacement (down), doing negative work since the block now has less gravitational potential energy.
Power
Power is the rate of transfer of energy, often the rate of doing work. Power can be calculated by the equation P = W/t. Power is measured by Watts (W), where 1 W = 1 J/s. You might be familiar with another unit of power: horsepower (about 745 Watts). Misleadingly, a horse can generate up to 15 horsepower and not 1.
Relating Energy/Work/Power to Forces and Motion
To relate everything together, you can do a simple experiment at home to measure how "powerful" you are.
In this experiment, you will perform several exercises and determine the force you apply when doing each exercise, determine the distance you are applying that force, and record the time taken. A sample calculation is "You lift a 5kg mass by 0.35 m 20 times in 30s"
The force you need to apply to the mass is to overcome the force of gravity on the object, which is 5g and around 50N
The total distance the mass moves is 0.35m * 20 times = 7 meters
Total work done is 50*7 = 350J
And your average power is 350/30s = 11.7 W
One exercise can be stair climbs. When you climb a flight of stairs, the force you apply is equal to your weight. You need to first change weight in pounds to newton, 1lb = 4.45N. The distance you apply this force is the height of the flight of stairs. After you measure your weight, height of the stairs, and the time it takes for you to climb those stairs, we can calculate. Here are the calculations I have:
Force = 140 * 4.45 = 623N
Distance = 2.11m
Time = 1.78s
Work = 623 * 2.11 = 1314.53
Power = 1314.53/1.78 = 738.5 W, which is about 1 horsepower.
Another exercise can be bicep curls. The force you apply is the weight of the dumbbell. The distance you apply this force is about your arm length. You can do multiple repetitions, but make sure to multiply your arm length by the number of reps to get the total distance travelled. Here are my calculations:
Force = 5lb * 4.45 = 22.2N
Reps = 5
Total Distance = 0.67m (arm length) * 5 = 3.35m
Time = 4.87s
Work = 22.2 * 3.35 = 74.37J
Power = 74.37/4.87 = 15.3 W, which is a lot less than the power I generated when climbing stairs. This makes sense because when you climb stairs, you use a lot of big muscles in your legs which can generate more energy in a shorter period of time compared to the small biceps muscles.
Energy, work, power, forces and motion are all related. From the previous units, we saw how forces and motion are related. Now we know how forces and work is related (W = F . x), work is a way to transfer energy and power is the rate of this transfer.
In this experiment, you will perform several exercises and determine the force you apply when doing each exercise, determine the distance you are applying that force, and record the time taken. A sample calculation is "You lift a 5kg mass by 0.35 m 20 times in 30s"
The force you need to apply to the mass is to overcome the force of gravity on the object, which is 5g and around 50N
The total distance the mass moves is 0.35m * 20 times = 7 meters
Total work done is 50*7 = 350J
And your average power is 350/30s = 11.7 W
One exercise can be stair climbs. When you climb a flight of stairs, the force you apply is equal to your weight. You need to first change weight in pounds to newton, 1lb = 4.45N. The distance you apply this force is the height of the flight of stairs. After you measure your weight, height of the stairs, and the time it takes for you to climb those stairs, we can calculate. Here are the calculations I have:
Force = 140 * 4.45 = 623N
Distance = 2.11m
Time = 1.78s
Work = 623 * 2.11 = 1314.53
Power = 1314.53/1.78 = 738.5 W, which is about 1 horsepower.
Another exercise can be bicep curls. The force you apply is the weight of the dumbbell. The distance you apply this force is about your arm length. You can do multiple repetitions, but make sure to multiply your arm length by the number of reps to get the total distance travelled. Here are my calculations:
Force = 5lb * 4.45 = 22.2N
Reps = 5
Total Distance = 0.67m (arm length) * 5 = 3.35m
Time = 4.87s
Work = 22.2 * 3.35 = 74.37J
Power = 74.37/4.87 = 15.3 W, which is a lot less than the power I generated when climbing stairs. This makes sense because when you climb stairs, you use a lot of big muscles in your legs which can generate more energy in a shorter period of time compared to the small biceps muscles.
Energy, work, power, forces and motion are all related. From the previous units, we saw how forces and motion are related. Now we know how forces and work is related (W = F . x), work is a way to transfer energy and power is the rate of this transfer.