Chapter 7 Work and Kinetic Energy. Copyright 2010 Pearson Education, Inc.

Transcription

1 Chapter 7 Work and Kinetic Energy

2 Units of Chapter 7 Work Done by a Constant Force Kinetic Energy and the Work-Energy Theorem Work Done by a Variable Force Power

3 7-1 Work Done by a Constant Force The definition of work, when the force is parallel to the displacement: SI unit: newton-meter (N m) = joule, J (7-1)

4 7-1 Work Done by a Constant Force

5 7-1 Work Done by a Constant Force If the force is at an angle to the displacement: (7-3)

6 7-1 Work Done by a Constant Force The work can also be written as the dot product of the force and the displacement:

7 7-1 Work Done by a Constant Force The work done may be positive, zero, or negative, depending on the angle between the force and the displacement:

8 7-1 Work Done by a Constant Force If there is more than one force acting on an object, we can find the work done by each force, and also the work done by the net force: (7-5)

9 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: y Δr x w W g = w! y cos180 = " mg! y

10 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 An FBD for the box: x F N y θ θ w Apply Newton s 2 nd Law: " " F F x y = = F N! wsin# = 0! wcos# = 0

11 Example continued: The magnitude of F is: F = mg sin! If the box travels along the ramp a distance of Δx the work by the force F is W F = F" x cos0 = mg" xsin! The work by gravity is (! + 90 ) = # mg xsin! W g = w" x cos "

12 Example continued: The work by the normal force is: W N = N! x cos 90 = 0 The net work done on the box is: W net = W F + W g + W = mg! xsin# " mg! xsin# + = 0 N 0

13 Example: What is the net work done on the box in the previous example if the box is not pulled at constant speed? # F ma F = F x + = " wsin! = ma wsin! Proceeding as before: W net = W = F + W g + W N ( ma + mg # ) sin! x " mg! xsin# + 0 = ( ma)! x = F! x net

14 7-2 Kinetic Energy and the Work-Energy Theorem When positive work is done on an object, its speed increases; when negative work is done, its speed decreases.

15 7-2 Kinetic Energy and the Work-Energy Theorem After algebraic manipulations of the equations of motion, we find: Therefore, we define the kinetic energy: (7-6)

16 Example: The extinction of the dinosaurs and the majority of species on Earth in the Cretaceous Period (65 Myr ago) is thought to have been caused by an asteroid striking the Earth near the Yucatan Peninsula. The resulting ejecta caused widespread global climate change. If the mass of the asteroid was kg (diameter in the range of 4-9 miles) and had a speed of 30.0 km/sec, what was the asteroid s kinetic energy? 1 2 K = mv = 2 = 4.5! J ( 16 )( 3 10 kg m/s) 2! This is equivalent to ~10 9 Megatons of TNT.

17 7-2 Kinetic Energy and the Work-Energy Theorem Work-Energy Theorem: The total work done on an object is equal to its change in kinetic energy. (7-7)

18 7-3 Work Done by a Variable Force If the force is constant, we can interpret the work done graphically:

19 7-3 Work Done by a Variable Force If the force takes on several successive constant values:

20 7-3 Work Done by a Variable Force We can then approximate a continuously varying force by a succession of constant values.

21 7-3 Work Done by a Variable Force The force needed to stretch a spring an amount x is F = kx. Therefore, the work done in stretching the spring is (7-8)

22 Example: (a) If forces of 5.0 N applied to each end of a spring cause the spring to stretch 3.5 cm from its relaxed length, how far does a force of 7.0 N cause the same spring to stretch? F = F 1 1 For springs F x. This allows us to write. 2 x x 2 F & 7.0 N # $ % 5.0 N 2 Solving for x 2 : x = x = ( 3.5 cm) 4.9 cm. 2 1! = F1 "

23 Example continued: (b) What is the spring constant of this spring? k = x F N 3.5 cm 1 = = 1.43 N/cm. Or k = x F N 4.9 cm 2 = = 1.43 N/cm.

24 Example: An ideal spring has k = 20.0 N/m. What is the amount of work done (by an external agent) to stretch the spring 0.40 m from its relaxed length? F x (N) kx 1 x 1 =0.4 m x (m) W = = Area under curve ( kx )( x ) = kx = ( 20.0 N/m)( 0.4 m) 1.6 J =

25 7-4 Power Power is a measure of the rate at which work is done: (7-10) SI unit: J/s = watt, W 1 horsepower = 1 hp = 746 W

26 7-4 Power

27 7-4 Power If an object is moving at a constant speed in the face of friction, gravity, air resistance, and so forth, the power exerted by the driving force can be written: (7-13)

28 Example: A race car with a mass of kg completes a quarter-mile (402 m) race in a time of 4.2 s starting from rest. The car s final speed is 125 m/s. What is the engine s average power output? Neglect friction and air resistance. P av = = " E " t " K " t = = " U + " K " t 1 2 mv " t 2 f = 9.3! 10 5 watts

29 Summary of Chapter 7 If the force is constant and parallel to the displacement, work is force times distance If the force is not parallel to the displacement, The total work is the work done by the net force:

30 Summary of Chapter 7 SI unit of work: the joule, J Total work is equal to the change in kinetic energy: where

31 Summary of Chapter 7 Work done by a spring force: Power is the rate at which work is done: SI unit of power: the watt, W