Episode 13 explains why the law of conservation of energy is better stated as “the total mechanical energy of a system is conserved unless net external work is applied to the system.”
Episode 13 explains why the law of conservation of energy is better stated as “the total mechanical energy of a system is conserved unless net external work is applied to the system.” Beginning with an explanation of mechanical energy (1:51) before skating into two practice problems (3:08). Energy is conserved unless there is net external work (7:07). Don’t forget that work can be negative too if it takes energy away from a system. (9:03) The episode concludes with an introduction to power (10:19).
The Question of the Day asks: (11:16)
A frictionless lab cart is released from rest on a ramp from vertical height “h” and a ball is dropped from rest at the same vertical height. Which is moving faster the moment before impacting the ground?
a) the cart
b) the ball
c) they are moving with the same speed
Thank you for listening to The APsolute RecAP: Physics 1 Edition!
(AP is a registered trademark of the College Board and is not affiliated with The APsolute RecAP. Copyright 2020 - The APsolute RecAP, LLC. All rights reserved.)
Website:
EMAIL:
Follow Us:
Hi and welcome to the APsolute RecAP: Physics 1 Edition. Today’s episode will focus on the broad topic of Energy.
Let’s Zoom out:
Unit 4 – Energy
Topic 4.1-4.3
Big Idea – Force Interactions, Change, and Conservation
While studying dynamics, you were introduced to solving problems on inclines. So, it was possible to draw a free-body diagram for an object on a ramp, sum the forces and set equal to m*a. But… what if that ramp was replaced with a semi-circular hill like a halfpipe in skateboarding or a rolling hill like on a ski slope? The angle of those inclines change along their path, and it would be painful to calculate the acceleration and then the new velocity for each little bit. Enter work and energy!
Let’s Zoom in:
First and foremost, this is a good time to grab a Physics 1 equation sheet or at least open the PDF on your phone because there are quite a few formulas in this unit (6 to be exact). It is important to understand that all of these equations stem from a unifying concept, the Law of Conservation of Energy. “Energy is neither created nor destroyed!” … kind of. That gives people the impression that energy cannot increase or decrease which just plain isn’t true. The law of conservation of energy is better stated as the total mechanical energy of a system is conserved unless net external work is applied to the system.
So what is mechanical energy? Well, for Physics 1, it is gravitational potential energy, kinetic energy, and elastic potential energy. The formula sheet refers to these as Ug, K, and Us respectively. Circle these equations on the formula sheet now (2 second pause), no… really, circle them. Gravitational potential energy changes if an object has mass and is displaced in a gravitational field, little “g”. Meaning, if the object is lifted or falls downward, it has changed its gravitational potential energy. Kinetic energy is possessed by a system if it has mass and speed. If the object is moving, it has kinetic energy. Elastic potential energy is possessed by a spring if it is stretched or compressed a certain displacement. Additionally, each spring is unique and has a constant (k) associated with how much force is required to stretch or compress the spring a certain distance. Those three forms of mechanical energy need to be determined and calculated at each position to determine the total energy of the system. If the total mechanical energy increases or decreases throughout a process, then we say that work was done on the system to cause the energy change.
Let’s try an easy one. A 50kg skateboarder is at rest at the top of a 5 m tall halfpipe. How fast will the skateboarder be going at the bottom? For this question, friction is negligible. We should check what forms of energy the skateboarder possesses at the top of the ramp. They aren’t moving yet and there is no spring, so no kinetic or elastic potential energies. They are located at a height above their lowest points so we would say that they possess 50kg * 10 m/s2* 5 m = 2500 Joules. Wait… what’s a Joule? It is the unit for work and energy. Simply put, it is a kilogram meter squared over a second squared. It’s a mouthful is what it is! So, scientists renamed it a Joule. So, all 2500J will be there in the beginning, and will still be there in the end since it is frictionless. At the bottom of the halfpipe, the skater will no longer have a height of 5 m, but they will have a considerable amount of kinetic energy. Setting 2500 J equal to ½ m*v2and doing some math, you find they are moving with a speed of 10 m/s. Simple enough.
As a kid, I liked watching Wile E Coyote unsuccessfully chase after the road runner. You know… Meep Meep. On one particular episode I recall the coyote was on roller skates and had compressed a spring a distance that looked to be about 5 meters or so (I’m estimating of course). The coyote was trying to use the energy in the compressed spring to propel himself horizontally and then up a vertical cliff wall much like the halfpipe in the previous problem. The cliff wall was approximately 30 m in vertical height, and we will assume that Wile E Coyote has a mass of 20 kg. What spring constant should the cartoon carnivore be shopping for at the Acme Spring Store? Well, at the start, the coyote-spring system will have elastic potential energy stored in the compressed spring. After launching off of the spring, his energy will be in the form of kinetic energy. Then he will gradually climb up the cliff wall and the kinetic energy will be transformed into gravitational potential energy. We can actually figure out how much energy is required by multiplying the mass (20 kg), the acceleration due to gravity (10 m/s^2) and the cliff height (30 m) to see that he needs to store 6000 J of energy in the spring before launching. Setting the total mechanical energy equal to the elastic potential energy at the start, we can determine the spring constant (k) is equal to 480 kg/s^2 or 480 N/m. I know… weird units right? Remember Newtons are (kg*m) / s^2, so dividing by meters yields kilograms per second squared.
So, you might find yourself asking, well isn’t total mechanical energy always constant then? The short answer is “no”. The longer answer is, “Energy is conserved unless there is net external work.” What does that even mean? In the most simple sense, work is an energy change. So, a bobsled that sits at rest on the icy starting line of a level bit of track has no energy initially until work is done by an applied force by the pushers. Let’s say they push with a combined force of 1000 N for a distance of 20 m. Well, the work they do on the bobsled is equal to the force multiplied by the displacement multiplied by the cosine of the angle between the force and displacement. In this case, that angle is 0 degrees so cosine 0 degrees is positive 1. So, the work they did is 20,000 J. Since work is a measure of energy change and they seem to gain kinetic energy, we can actually figure out the speed of the bobsled team if we know their combined mass with the sled is 500 kg. We can set work equal to the final kinetic energy. I am doing this because I am always going back to the fundamental law of the universe… total mechanical energy initial + net work = total mechanical energy final. Since the sled had no initial mechanical energy then the work done by the friction with the ice is equal to the total mechanical energy final or in our case the kinetic energy. 20,000 J equals ½ m v^2 so the final speed would be about 9 m/s. If you graphed the continuous force of 1000N vs the displacement of 20 m you would notice it forms a rectangle with an area equal to… 20,000 J! Would you look at that!
So, I will tell you - That is a HUGE time saver on multiple choice questions. Knowing that the area of a force vs displacement graph represents the work done on a system will save you a ton of headaches, trust me.
So, is work always positive? Does it always add energy to the system? No. Work can be negative too, it takes energy away from a system. Take a shuffleboard puck for example. When released, it has a velocity so it has kinetic energy. Eventually it is brought to rest by negative work done by friction. That is because friction acts at 180 degrees to the forward displacement of the puck. In fact, you could figure out the coefficient of friction if you know how fast the puck was initially moving and how long it traveled before stopping. If the ½ kg shuffleboard puck is released with a speed of 8 m/s and slides 10 m before coming to rest, then it initially had 16 J of kinetic energy that is loses along the court. Since friction is the only force acting over the entire 10 m distance at 180 degrees we see that the work done by friction is -16 J and the force of kinetic friction must be 1.6 N. Remembering all your work on dynamics you will recall that the coefficient of kinetic friction (mu) is equal to kinetic friction divided by the normal force, in this case m*g. So, the coefficient of kinetic friction is 0.32.
Finally, we should probably at least mention power. It is the rate at which work is done. So you shouldn’t be surprised that the equation for power is work over time or energy change over time and that the units are J/s. And… now, you know what? Watt! No, really… a J/s is called a Watt, w-a-t-t. Abbreviated capital “W”.
To Recap…
Total mechanical energy is the combination of gravitational potential energy, kinetic energy, and elastic potential energy. Change in total mechanical energy of a system is equal to the net work done on the system. Or, Total ME initial + work = total ME final. Work is positive when the force is in the direction of displacement, and negative when the force is in the direction opposite the displacement. Power is the rate of energy change over time.
Coming up next on the APsolute RecAP Physics 1 Edition, we will look in depth at the spring constant and Hooke’s Law.
Today’s Question of the Day focuses on work and energy.
Question: A frictionless lab cart is released from rest on a ramp from vertical height “h” and a ball is dropped from rest at the same vertical height. Which is moving faster the moment before impacting the ground?
a) the cart
b) the ball
c) they are moving with the same speed