Kinetic and gravitational potential energy. Let's consider a dropped ball again. As I release it, its velocity is zero. As it gets to the floor, it's attained a finite velocity. From what we know already, we can figure out the relationship between the height h from which it was dropped and the speed v, which it had when it hit the floor: distance fallen h = average speed x time=(1/2)vxtime v=time x g Therefore time =v/g. Put it in the 1st equation: h=(1/2)v(v/g) or gh=(1/2)v2

Kinetic and gravitational potential energy. From F=ma and the fact that the gravitational force is mg, we thus conclude that, for the dropped ball: gh=(1/2)v2 or multiplying both sides by m mgh=(1/2)mv2 at start just before it hits We say that the ball had Gravitational potential energy =mgh At the start of its fall and that energy changed form and became Kinetic energy = (1/2)mv2 Just before the ball hit the floor

Conservation of Energy In fact, a more general analysis shows that at any moment during the fall, the sum Gravitational energy + kinetic energy stays the same. At the beginning of the fall, the energy is all gravitational. At the bottom of the fall it's all kinetic. In between, a gradual change of form of the energy takes place, but the total amount of energy stays the same. This is what happens to the energy of water going over falls or a dam. In hydropower systems, the energy is stored in the water above the dam, and then converted to kinetic energy as it goes over the dam. The kinetic energy in the water is then used to do useful things like generate electricity as we will discuss later.

Hoover Dam on the Colorado river In Nevada Height of the water at the top above the bottom= 180m Volume of water in The lake above the Dam= 32 km3

Review velocity is (change in position)/time elapsed acceleration is (change in velocity/time elapsed total force =mass x acceleration gravitational force = Mass x g down From F=ma for falling object (1/2)mv2 + mgh = constant kinetic gravitational energy potential energy Conservation of energy when the only forces acting to accelerate the body are gravitational. When other forces are acting there are more terms on the left hand side.

If I dropped a ball from a height of 2 meters, what was its speed just before it hit the floor? A. 9.8x2 m/s B. 9.8/2 m/s _____ C. √9.8x2 m/s D. √9.8x4 m/s E. 9.8x4 m/s

Units of force: Since total force is mass x acceleration a suitable metric unit is 1kg meter/second2 called 1 Newton In the British system, the force unit chosen is the pound. The British mass unit (not used much) is a slug and 1 pound = 1slug ft/second2 1 Newton = .225 pounds We will tell you this conversion value if you need it. (Not in Appendix B).

Weight: The magnitude of the gravitational force on an object is called its weight. Therefore the weight of an object of mass M is always Mg near the surface of the earth. Because g varies with altitude, the weight can be different in different places for a given object but the mass does not change. The units of weight are therefore force units.

Units of energy: We can figure out the units of energy from the expression for the gravitational potential energy: mgh= (a force) times (a length)‏ So metric units of force are Newtonsxmeters which we call joules. 1 joule = 1 newton meter British units of energy can then be 1 ft-pound Confusingly, this is NOT a British Thermal Unit (btu)‏ 1Btu= 1055Joules=778 ft-lb

Again, you do not have to remember the conversion factors which are listed in Chapter 3, Appendix B and inside the back cover of your book. However you are expected to be able to use the conversion factors to change a quantity from one kind of units to another. It is also important to always use the right KIND of units: energy units for energy, force units for force, etc.

Energy transfers between objects. Consider holding up the book again. Suppose its mass is 1kg and that I slowly raise it 1 meter. How much gravitational potential energy did it gain? A. 1 joule B. 9.8 joules C. 1 newton D. 9.8 newtons

OK it gained 9.8 joules. Since energy is conserved, where did this energy come from? A. The kinetic energy of the book. B. chemical energy in my body C. the action of the friction of the air. D. chemical energy in the book.

Now I want to concentrate on how the energy was transferred from my body to the book. I pressed my hand against the bottom of the book, and produced an upward force which was just slightly larger than the downward gravitational force mg so that the book slowly moved up a distance h. In magnitude the resulting gain in energy was mg x h which is equal to (the upward force with which I pushed with my hand)‏ times (the distance through which I pushed the book along the direction of the force. )‏

This is an example of the transfer of energy from one object to another through the performance of WORK. When one object exerts a force F on another and the second object moves along the direction of the force a distance d, then we say that the first object has done work Fd on the second object, resulting in energy transfer Fd from the first to the second object.

Notes about work: Work has the units of energy but it is always an amount of energy transferred from one object to another and NOT the amount of energy in a object. After the energy is transferred through the performance of work, the energy ends up in the second object in one of the forms we will be discussing such as gravitational potential, kinetic, thermal, chemical or electrical potential energy. In the formula work=Fd, d is the distance ALONG THE FORCE. This may not be the same as the distance which the contact point between one object and the other moves.

Suppose you slowly carry an object weighing 10 pounds up a flight of stairs consisting of 20 steps. Each step is 1/2 feet high and 1 foot wide. How much work did you perform on the object? A. 100 ft-pounds B. 200 ft-pounds C. (5)1/2x100 ft-pounds D. 0

You carry the same 10 pound object 100 ft down the horizontal hall. During the time between the time after you picked it up and before you put it down, how much work did you perform on the object? (Suppose you walked slowly at constant velocity). A. 1000 ft-pounds B. 0 C. 500 ft-pounds D. insufficient information is given

From a zero velocity start, I push a cart along a frictionless track with a constant force for 1/2 meter and then let it go. Its velocity after I release it is observed to be 1m/s and the mass of the cart is 1/4 kg. What was the magnitude of the force with which I pushed it? A. 0.25N B. 2.45N C. 0.5N D. 0 Question 6

I throw a 1/2 kg object straight up. During the throw, I exert a constant force upward on the object while it moves upward through 1/2 meter. Just after I release the object, it is observed to have an upward velocity of magnitude (speed) 2 meters per second. What was the magnitude of the force which I exerted on the object during the throw? A. 4.8 N B. 6.9 N C. 0.25N D. 0.5N Question 7

Answer: B The point here is that the object ends up with two forms of additional energy, kinetic and gravitational potential: Fd = (1/2)mv2 +mgd Work gain in gain in gravitational Done kinetic potential energy energy F(1/2 m)=(1/2)(1/2kg)(2m/s)2 +(1/2kg)(9.8m/s2)(1/2 m) Divide both sides by ½ meter and simplify F=2kgm/s2 + 4.9 kgm/s2 = 6.9 newtons

OK that was the force exerted by my hand on the ball. What was the total force on the ball during the toss? (1/2 kg ball, pushed up 1/2m, final speed 2 m/s). A. 6.9 newtons B. 4.9 newtons C. 2 newtons D. 0 Question 11

The total force on the ball was 2N upward and the velocity of the ball was 2m/s as it left my hand. The mass of the ball was 1/4 kg. How long (how many seconds) was a pushing the ball upward before it left my hand? A. 0.25 seconds B. 0.5 seconds C. 4.9 seconds D. 9.8 seconds E. .204 seconds

Summary: Energy is transferred from one object to another through the performance of work. The amount of work done by one object on another is Work done = F x d where F is the force which the first object exerts on the second and d is the distance, along the direction of the force, through which the force acts. The work done is the amount of energy which is transferred from the first object to the second. The energy ends up in one of the forms we are discussing: kinetic, gravitational potential, thermal. It is not meaningful to talk of work IN a body. If the force acting is opposite in direction to the direction through which it acts, the work done is negative and energy is extracted from the second object.