Energy [J] is defined as the ability to do work. Or Work is the Energy supplied to on object to make it move.

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1 Work and Energy Work (W) is done on an object by an force when the object moves through a distance (displacement). Since Force and displacement are vectors, work has to be a scalar. We use the scalar product: W r r = F s = Fs cosθ [Joule J = Nm] Energy [J] is defined as the ability to do work. Or Work is the Energy supplied to on object to make it move. Work is an energy transfer by the application of a force. For work to be done there must be a nonzero displacement. How much work is 1 Joule? Let s compare Annual U.S. energy use Mt. St. Helens eruption Burning one gallon of gas Human food intake/day Melting an ice cube Lighting a 100-W bulb for 1 minute Heartbeat Turning page of a book Hop of a flea Breaking a bond in DNA 8 x

2 The Law of Conservation of Energy The total energy of the Universe is unchanged by any physical process. The three kinds of energy are: kinetic energy, potential energy, and rest energy. Energy may be converted from one form to another or transferred between bodies.

3 Work done by a force an example Consider you are pushing a box, then: More work is required to exert a greater force for a finite distance More work is required to exert a finite force over a greater distance Therefore: Work equals force times displacement r r W = F s = Fs cosθ [Joule J = Nm] θ = 0 ; cos0 = 1(angle between the two vectors) W = F s

4 Work W can be. +W 0 -W Example 1: A person pulls a suitcase If the person would pull horizontal little force would be necessary to do the same work!!! Example 2: Work done by gravity Force needed to lift up the box F 1 = mg, W 1 =Fs=(mg)h =mgh cos(0)=mgh Work done by gravity W g = mgh cos(180)= -mgh

5 It is only the force in the direction of the displacement that does work. Free Body Diagram for the box F y θ Δr x N Δr x θ x w F W F ( F cos ) Δx = F Δr = θ x x The work done by the force N is: W N = 0 The normal force is perpendicular to the displacement. The work done by gravity (w) is: W g = 0 The force of gravity is perpendicular to the displacement. W net = W F + W N + W g = ( F cosθ ) Δx = ( F cosθ ) Δx

6 Example: A ball is tossed straight up. What is the work done by the force of gravity on the ball as it rises? FBD for rising ball: W g y w = mgδy Δr x r r = WΔy cos180

7 Conceptual Checkpoint Which way is more work?

8 Total Work?? When more then a force acts on an object the total work. Force F 1 (e.g. Friction) does work W 1, Force F 2 does work W 2, etc. W 3 total = W1 + W2 + W +... = W OR Calculate the net force (or total force) W = r r F s = total F total i s cosθ

9 Example: A box of mass m is towed up a frictionless incline at constant speed. The applied force F is parallel to the incline. What is the net work done on the box? F y x F N θ θ Apply Newton s 2 nd Law: F F x y = F wsinθ = 0 = N The magnitude of F is: wcosθ = 0 w F = mgsinθ If the box travels along the ramp a distance of Δx the work by the force F is W F = FΔx cos0 = mgδ xsinθ The work by gravity is Wg = w Δ x cos( θ + 90 ) = mg Δ xsinθ

10 Example continued: The work by the normal force is: W N = NΔx cos 90 = 0 The net work done on the box is: W net = W F + W g + W N = mgδxsinθ mgδxsinθ + 0 = 0

11 Graphical Representation of Work Plot force vs position and the area under the curve represents the Work W= F d (The area of a rectangle with length a and with b: Area = ab)

12 What if Force isn t constant? Split the curve in several intervals/ rectangles and add them up OR with calculus

13 Are work and speed are related? Of course: When the total work done on a object is: Positive, its speed increases (W >0, v f >v i ) Negative, ist speed decreases (W<0, v f <v i ) Conclusion: There is a connection between work and change in speed

14 Work-Energy Theorem Kinetic Energy (K or E k ) is energy associated with the state of motion of an object. The faster the object moves, the greater is its kinetic energy. When the object is stationary, its kinetic energy is zero. Kinetic Energy: KE=E kin = ½ mv 2 linear KE=E rot = ½ Jω 2 rotational Net work that is done on body (by net force) equals change in kinetic energy W net = ΔKE=E kin = ½ mv 22 -½ mv 1 2 all moving bodies (v 0 if linear, ω 0 if rotational) If we see something with mass moving, we know is has kinetic energy!!!!!

15 Remember Springs? What about the work done on a spring? r r Hooke s Law F = k Δx (Hooke s Law) F is the magnitude of the force exerted by the free end of the spring, x is the measured stretch of the spring, and k is the spring constant (a constant of proportionality; its units are N/m). Remember: F ~ x Area: W=Fd W=(kx)x/2 W= kx 2 /2

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17 Power Power is a measure how quickly work is done P=W/t or P= Fs/t=F(s/t)=Fv SI Unit J/s= W (watt) Other common unit Horsepower hp: 1hp = 746W Source Hoover Dam Car moving at 40 mph Home stove Sunlight falling on one square meter Refrigerator Television Person walking up stairs Human brain Approximate power (W) 1.34 x x x

18 Example You want to accelerate your car from 13.4m/s (30mph) to 17.9m/s (40mph). What is the minimum power your car need to do this procedure?

19 Calculate the power output of a 1.8-g beetle as it walks up a window pane at 2.3 cm/s. The beetle walks on a path that is at 25 to the vertical, as illustrated in Figure 7 18.

20 Two springs, with force constants and are connected in parallel, as shown in Figure. How much work is needed to stretch this system a distance x?