1. CHAPTER 3 KINETICS OF A PARTICLE: WORK AND ENERGY

2. 3.1 The Work of a Force Today’s Objectives: Students will be able to: a)Calculate the work of a force. b)Apply the principle of work and energy to a particle or system of particles. In-Class Activities: Check homework, if any Applications Work of a force Principle of work and energy

3. APPLICATIONS A roller coaster makes use of gravitational forces to assist the cars in reaching high speeds in the “valleys” of the track. How can we design the track (e.g., the height, h, and the radius of curvature,  ) to control the forces experienced by the passengers?

4. APPLICATIONS (continued) Crash barrels are often used along roadways for crash protection. The barrels absorb the car’s kinetic energy by deforming. If we know the typical velocity of an oncoming car and the amount of energy that can be absorbed by each barrel, how can we design a crash cushion?

5. WORK AND ENERGY Another equation for working kinetics problems involving particles can be derived by integrating the equation of motion (F = ma) with respect to displacement. This principle is useful for solving problems that involve force, velocity, and displacement. It can also be used to explore the concept of power. By substituting a t = v (dv/ds) into F t = ma t, the result is integrated to yield an equation known as the principle of work and energy. To use this principle, we must first understand how to calculate the work of a force.

6. WORK OF A FORCE A force does work on a particle when the particle undergoes a displacement along the line of action of the force. Work is defined as the product of force and displacement components acting in the same direction. So, if the angle between the force and displacement vector is , the increment of work dU done by the force is dU = F ds cos  By using the definition of the dot product and integrating, the total work can be written as  r2r2 r1r1 U 1-2 = F dr

7. WORK OF A FORCE (continued) Work is positive if the force and the movement are in the same direction. If they are opposing, then the work is negative. If the force and the displacement directions are perpendicular, the work is zero. If F is a function of position (a common case) this becomes   s2s2 s1s1 F cos  ds U 1-2 If both F and  are constant (F = F c ), this equation further simplifies to U 1-2 = F c cos  s 2 - s 1 )

8. WORK OF A WEIGHT The work done by the gravitational force acting on a particle (or weight of an object) can be calculated by using The work of a weight is the product of the magnitude of the particle’s weight and its vertical displacement. If  y is upward, the work is negative since the weight force always acts downward. - W (y 2 - y 1 ) = - W  y - W dy = U 1-2 =  y2y2 y1y1

9. WORK OF A SPRING FORCE When stretched, a linear elastic spring develops a force of magnitude F s = ks, where k is the spring stiffness and s is the displacement from the unstretched position. If a particle is attached to the spring, the force F s exerted on the particle is opposite to that exerted on the spring. Thus, the work done on the particle by the spring force will be negative or U 1-2 = – [ 0.5k (s 2 ) 2 – 0.5k (s 1 ) 2 ]. The work of the spring force moving from position s 1 to position s 2 is = 0.5k(s 2 ) 2 - 0.5k(s 1 ) 2 k s ds F s dsU 1-2 s2s2 s1s1 s2s2 s1s1  

10. SPRING FORCES 1.The equations just shown are for linear springs only! Recall that a linear spring develops a force according to F = ks (essentially the equation of a line). 3.Always double check the sign of the spring work after calculating it. It is positive work if the force put on the object by the spring and the movement are in the same direction. 2.The work of a spring is not just spring force times distance at some point, i.e., (ks i )(s i ). Beware, this is a trap that students often fall into! It is important to note the following about spring forces:

11. 3.2 Principle of Work and Energy  U 1-2 is the work done by all the forces acting on the particle as it moves from point 1 to point 2. Work can be either a positive or negative scalar. By integrating the equation of motion,  F t = ma t = mv(dv/ds), the principle of work and energy can be written as  U 1-2 = 0.5m(v 2 ) 2 – 0.5m(v 1 ) 2 or T 1 +  U 1-2 = T 2 T 1 and T 2 are the kinetic energies of the particle at the initial and final position, respectively. Thus, T 1 = 0.5 m (v 1 ) 2 and T 2 = 0.5 m (v 2 ) 2. The kinetic energy is always a positive scalar (velocity is squared!). So, the particle’s initial kinetic energy plus the work done by all the forces acting on the particle as it moves from its initial to final position is equal to the particle’s final kinetic energy.

12. PRINCIPLE OF WORK AND ENERGY (continued) The principle of work and energy cannot be used, in general, to determine forces directed normal to the path, since these forces do no work. Note that the principle of work and energy (T 1 +  U 1-2 = T 2 ) is not a vector equation! Each term results in a scalar value. Both kinetic energy and work have the same units, that of energy! In the SI system, the unit for energy is called a joule (J), where 1 J = 1 N·m. In the FPS system, units are ft·lb. The principle of work and energy can also be applied to a system of particles by summing the kinetic energies of all particles in the system and the work due to all forces acting on the system.

13. 3.3 Principle of Work and Energy for a System of Particles The principle of work and energy can be extended to include a system of particles isolated within an enclosed region of space. Symbolically, the principle looks like  T 1 +  U 1-2 =  T 2 In works, this equations states that ~ The system’s initial kinetic energy (  T1) plus the work done by all the external and internal force acting on the particles of the system (  U1-2) is equal to the system’s final kinetic energy (  T2). Note that although the internal force on adjacent particles occur in equal but opposite collinear pairs, the total work done by each of these forces will, in general, not cancel out since the paths over which corresponding particles travel will be different. There are two important exceptions to this rule which often occur in practice. - When particles are contained within the boundary of a translating rigid body. - When particles are connected by inextensible cables. In this cases, adjacent particles exert equal but opposite internal forces that have components which undergo the same displacement, and therefore the work of these forces cancels.

14. Special class of problem involving work of friction caused by sliding. We note also that equation “  T1 +  U1-2 =  T2” can be applied to problems involved sliding friction; however it should be realized that the work of the resultant friction force is not represented by µkNS; instead, this term represents both the external work of friction (µkNS’) and internal work µkN(S – S’) which is converted into various forms of internal energy, such as heat.

15. 3.4 Power and Efficiency Today’s Objectives: Students will be able to: a)Determine the power generated by a machine, engine, or motor. b)Calculate the mechanical efficiency of a machine. In-Class Activities: Check homework, if any Applications Define power Define efficiency

16. APPLICATIONS Engines and motors are often rated in terms of their power output. The power requirements of the motor lifting this elevator depend on the vertical force F that acts on the elevator, causing it to move upwards. Given the desired lift velocity for the elevator, how can we determine the power requirement of the motor?

17. APPLICATIONS (continued) The speed at which a vehicle can climb a hill depends in part on the power output of the engine and the angle of inclination of the hill. For a given angle, how can we determine the speed of this jeep, knowing the power transmitted by the engine to the wheels?

18. POWER Thus, power is a scalar defined as the product of the force and velocity components acting in the same direction. Since the work can be expressed as dU = F dr, the power can be written P = dU/dt = (F dr)/dt = F (dr/dt) = F v If a machine or engine performs a certain amount of work, dU, within a given time interval, dt, the power generated can be calculated as P = dU/dt Power is defined as the amount of work performed per unit of time.

19. POWER (continued) Using scalar notation, power can be written P = F v = F v cos  where  is the angle between the force and velocity vectors. In the FPS system, power is usually expressed in units of horsepower (hp) where 1 hp = 550 (ft · lb)/s = 746 W. So if the velocity of a body acted on by a force F is known, the power can be determined by calculating the dot product or by multiplying force and velocity components. The unit of power in the SI system is the watt (W) where 1 W = 1 J/s = 1 (N ·m)/s.

20. EFFICIENCY If energy input and removal occur at the same time, efficiency may also be expressed in terms of the ratio of output energy to input energy or  = (energy output)/(energy input) Machines will always have frictional forces. Since frictional forces dissipate energy, additional power will be required to overcome these forces. Consequently, the efficiency of a machine is always less than 1. The mechanical efficiency of a machine is the ratio of the useful power produced (output power) to the power supplied to the machine (input power) or  = (power output)/(power input)

21. Solving Problems Multiply the force magnitude by the component of velocity acting in the direction of F to determine the power supplied to the body (P = F v cos  ). If the mechanical efficiency of a machine is known, either the power input or output can be determined. Determine the velocity of the point on the body at which the force is applied. Energy methods or the equation of motion and appropriate kinematic relations, may be necessary. In some cases, power may be found by calculating the work done per unit of time (P = dU/dt). Find the resultant external force acting on the body causing its motion. It may be necessary to draw a free-body diagram.

22. Given:A sports car has a mass of 2 Mg and an engine efficiency of  = 0.65. Moving forward, the wind creates a drag resistance on the car of F D = 1.2v 2 N, where v is the velocity in m/s. The car accelerates at 5 m/s 2, starting from rest. Find:The engine’s input power when t = 4 s. Plan:1) Draw a free body diagram of the car. 2)Apply the equation of motion and kinematic equations to find the car’s velocity at t = 4 s. 3)Determine the power required for this motion. 4)Use the engine’s efficiency to determine input power. EXAMPLE

23. EXAMPLE (continued) Solution: 1)Draw the FBD of the car. The drag force and weight are known forces. The normal force N c and frictional force F c represent the resultant forces of all four wheels. The frictional force between the wheels and road pushes the car forward. 2)The equation of motion can be applied in the x-direction, with a x = 5 m/s 2 : +  F x = ma x => F c – 1.2v 2 = (2000)(5) => F c = (10,000 + 1.2v 2 ) N

24. EXAMPLE (continued) 3)The constant acceleration equations can be used to determine the car’s velocity. 4)The power output of the car is calculated by multiplying the driving (frictional) force and the car’s velocity: 5)The power developed by the engine (prior to its frictional losses) is obtained using the efficiency equation. P i = P o /  = 209.6/0.65 = 322 kW P o = (F c )(v x ) = [10,000 + (1.2)(20) 2 ](20) = 209.6 kW v x = v xo + a x t = 0 + (5)(4) = 20 m/s

25. 3.5 Conservative Force And Potential Energy Today’s Objectives: Students will be able to: a)Understand the concept of conservative forces and determine the potential energy of such forces. b)Apply the principle of conservation of energy. In-Class Activities: Check homework, if any Applications Conservative force Potential energy Conservation of energy

26. APPLICATIONS The weight of the sacks resting on this platform causes potential energy to be stored in the supporting springs. If the sacks weigh 100 lb and the equivalent spring constant is k = 500 lb/ft, what is the energy stored in the springs?

27. APPLICATIONS (continued) When a ball of weight W is dropped (from rest) from a height h above the ground, the potential energy stored in the ball is converted to kinetic energy as the ball drops. What is the velocity of the ball when it hits the ground? Does the weight of the ball affect the final velocity?

28. CONSERVATIVE FORCE A force F is said to be conservative if the work done is independent of the path followed by the force acting on a particle as it moves from A to B. In other words, the work done by the force F in a closed path (i.e., from A to B and then back to A) equals zero. This means the work is conserved.  rF0  · d x y z A B F A conservative force depends only on the position of the particle, and is independent of its velocity or acceleration.

29. CONSERVATIVE FORCE (continued) A more rigorous definition of a conservative force makes use of a potential function (V) and partial differential calculus, as explained in the texts. However, even without the use of the these mathematical relationships, much can be understood and accomplished. The “conservative” potential energy of a particle/system is typically written using the potential function V. There are two major components to V commonly encountered in mechanical systems, the potential energy from gravity and the potential energy from springs or other elastic elements. V total = V gravity + V springs

30. POTENTIAL ENERGY Potential energy is a measure of the amount of work a conservative force will do when it changes position. In general, for any conservative force system, we can define the potential function (V) as a function of position. The work done by conservative forces as the particle moves equals the change in the value of the potential function (the sum of V gravity and V springs ). It is important to become familiar with the two types of potential energy and how to calculate their magnitudes.

31. POTENTIAL ENERGY DUE TO GRAVITY The potential function (formula) for a gravitational force, e.g., weight (W = mg), is the force multiplied by its elevation from a datum. The datum can be defined at any convenient location. + W yV g  _ V g is positive if y is above the datum and negative if y is below the datum. Remember, YOU get to set the datum.

32. ELASTIC POTENTIAL ENERGY Recall that the force of an elastic spring is F = ks. It is important to realize that the potential energy of a spring, while it looks similar, is a different formula. Notice that the potential function V e always yields positive energy. 2 2 1 ksV e  V e (where ‘e’ denotes an elastic spring) has the distance “s” raised to a power (the result of an integration) or

33. 3.6 Conservation of Energy When a particle is acted upon by a system of conservative forces, the work done by these forces is conserved and the sum of kinetic energy and potential energy remains constant. In other words, as the particle moves, kinetic energy is converted to potential energy and vice versa. This principle is called the principle of conservation of energy and is expressed as 2211 VTVT  = Constant T 1 stands for the kinetic energy at state 1 and V 1 is the potential energy function for state 1. T 2 and V 2 represent these energy states at state 2. Recall, the kinetic energy is defined as T = ½ mv 2.

34. EXAMPLE Given: The girl and bicycle weigh 125 lbs. She moves from point A to B. Find:The velocity and the normal force at B if the velocity at A is 10 ft/s and she stops pedaling at A. Plan:Note that only kinetic energy and potential energy due to gravity (V g ) are involved. Determine the velocity at B using the conservation of energy equation and then apply equilibrium equations to find the normal force.

35. EXAMPLE (continued) Placing the datum at B: BBAA VTVT  22 ) 2.32 125 ( 2 1 )30(125)10)( 2.32 125 ( 2 1 B v  s ft V B 1.45  Equation of motion applied at B:  2 v mmaF nn  50 )1.45( 2.32 125 2  B N lbN B 283  Solution: