Work-Energy Theorem

1. Work-Energy Theorem • Recall that Work equals a change in the kinetic energy of an object ( W = ∆KE). • Therefore, W = KEafter - KEbefore • Also recall that W = F • d cosθ • And that KE = ½ mv2

2. Throwing a Baseball • The baseball before being thrown has zero velocity, therefore, its KEbefore = 0. • You add work to the baseball to get it moving, therefore, W > 0. • The baseball after being thrown has velocity and mass, therefore it has KE > 0. • This KE is equal to the initial W done. • KEbefore + W = KEafter

3. Catching a Baseball • The baseball before being caught has mass and velocity, therefore it has KE > 0. • The baseball after being caught has no velocity, therefore its KE = 0. • Therefore a work that is less than zero (W < 0) must have be done. • KEbefore + (-W) = KEafter

4. Practice Problem An 875.0-kg car speeds up from 22.0 m/s to 44.0 m/s while passing another car. What are its initial and final energies? How much work is done on the car to increase its speed?

5. Answer • KEinital = ½ (875 kg)(22 m/s)2 = 212000 J • KEfinal = ½ (875 kg)(44 m/s)2 = 847000 J • W = KEf - KEi = 847000 - 212000 = 635000 J

6. Gravitational Potential Energy • Potential energy can be thought of as stored energy. • PE = mgh • An object will have potential energy based upon the product of its mass, acceleration due to gravity, and the distance from a reference level. • Each of these different objects on the shelf have different PE based upon their masses and their distances from a reference level.

7. Remember Correct Signs! • Looking at this juggler it is important to remember that when the ball is going up, its displacement is upward, but the force of gravity (Fg) on the ball is downward. Hence, the work done by gravity is negative Wg = -mgh. • When the ball is going down the force and displacement are in the same direction. Hence the work done by gravity is positive Wg = +mgh.

8. Practice Problem A boy lifts a 2.2 kg book from his desk, which is 0.80 m high, to a bookshelf that is 2.10 m high. What is the potential energy of the book relative to the desk? What is the potential energy of the book relative to the ground?

9. Answer • PE = mgh = (2.2 kg)(9.8 m/s2)(2.1 m - 0.8 m) = 28 J • PE = mgh = (2.2 kg)(9.8 m/s2)(2.1 m) = 45.3 J

10. Law of Conservation of Energy • In a closed system, energy is neither created nor destroyed, rather it changes from one form of energy to another. The total energy of the system remains constant.

11. Mechanical Energy • The mechanical energy of a system is equal to the sum of the kinetic and potential energies (provided no other forms of energy are present). • ME = KE + PE

12. Conservation of Mechanical Energy • When mechanical energy is conserved, the sum of the kinetic and potential energies in a system before an event is equal to the sum of the kinetic and potential energies during and after the event. • KEbefore + PEbefore = KEafter + PEafter

13. Fill in the values for this event (remember the mechanical energy is conserved).

14. Answers • PE = (50 kg)(10 m/s2)(4 m) = 2000 J KE = ½ (50 kg)(0 m/s)2 = 0 J ME = 0 J + 2000 J = 2000 J v = 0 m/s • ME = still equals 2000 J PE = (50 kg)(10 m/s2)(3 m) = 1500 J KE = ME - PE = 2000 J – 1500 J = 500 J v = ? 3. Continue calculating!