Chapter 7 Work and Energy

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1 8/04/0 Lecture PowerPoints 009 Pearson Education, Inc. This work is protected by United States copyright laws and is provided solely for the use of instructors in teaching their courses and assessing student learning. Dissemination or sale of any part of this work (including on the World Wide Web) will destroy the integrity of the work and is not permitted. The work and materials from it should never be made available to students except by instructors using the accompanying text in their classes. All recipients of this work are expected to abide by these restrictions and to honor the intended pedagogical purposes and the needs of other instructors who rely on these materials. Copyright 009 Pearson Education, Inc. Chapter 7 Physics for Scientists and Engineers, with Modern Physics, 4 th Edition Giancoli Chapter 7 Work and Energy Copyright 009 Pearson Education, Inc.

2 8/04/0 Units of Chapter 7 Work Done by a Constant Force Scalar Product of Two Vectors Work Done by a Varying Force Kinetic Energy and the Work-Energy Principle Copyright 009 Pearson Education, Inc. 7- Work Done by a Constant Force The work done by a constant force is defined as the distance moved multiplied by the component of the force in the direction of displacement: Copyright 009 Pearson Education, Inc.

3 8/04/0 7- Work Done by a Constant Force In the SI system, the units of work is the joule: As long as this person does not lift or lower the bag of groceries, he is doing no work on it. The force he exerts has no component in the direction of motion. Copyright 009 Pearson Education, Inc. 7- Work Done by a Constant Force Example 7-: Work done on a crate. A person pulls a 50 kg crate 40 m along a horizontal floor by a constant force F P 00 N, which acts at a 37 angle as shown. The floor is smooth and exerts no friction force. Determine (a) the work done by each force acting on the crate, and (b) the net work done on the crate. Copyright 009 Pearson Education, Inc. 3

4 8/04/0 Example 7-: Work done on a crate. m 50 kg, F P 00 N, x 40 m (a) the work done by each force acting on the crate. (b) the net work done on the crate. r r (a) F and mg are perpendiculal to displacement (b) N so they do no work. r Work is done only by horizontal component of F W F xcosθ cos J P Since there is only one force doing work on crate the net work done on the crate 300 J P Copyright 009 Pearson Education, Inc. NB: This is NOT the same Example 7- Work done on a crate. As in text Giancoli. The example in Giancoli takes friction into account. m 50 kg, F P 00 N, F fr 50 N, x 40 m (a) the work done by each force acting on the crate. (b) the net work done on the crate. Copyright 009 Pearson Education, Inc. 4

5 8/04/0 NB: This is NOT the same Example 7- Work done on a crate. As in text Giancoli. The example in Giancoli takes friction into account. m 50 kg, F P 00 N, F fr 50 N, x 40 m (a) the work done by each force acting on the crate. (b) the net work done on the crate. r r (a) F and mg are perpendicular to displacement so they do no work. (b) N Work is done (i) by horizontal component of W F P x cosθ cos J and (ii) by the frictional force : W F fr x cosθ cos J Net work is calculated as J r F the sum of work done by all forces P Copyright 009 Pearson Education, Inc. 7- Work Done by a Constant Force Solving work problems:. Draw a free-body diagram.. Choose a coordinate system. 3. Apply Newton s laws to determine any unknown forces. 4. Find the work done by a specific force. 5. To find the net work, either a) find the net force and then find the work it does, or b) find the work done by each force and add. Copyright 009 Pearson Education, Inc. 5

6 8/04/0 7- Work Done by a Constant Force Example 7-: Work on a backpack. (a) Determine the work a hiker must do on a 5.0 kg backpack to carry it up a hill of height h 0.0 m, as shown. Determine also (b) the work done by gravity on the backpack, and (c) the net work done on the backpack. For simplicity, assume the motion is smooth and at constant velocity (i.e., acceleration is zero). Note that only the vertical component of the hiker s force need be considered: W Fdcosθ Fd.h/d Fh Copyright 009 Pearson Education, Inc. 7- Work Done by a Constant Force Example 7-: Work on a backpack. (a) Determine the work a hiker must do on a 5.0 kg backpack to carry it up a hill of height h 0.0 m, as shown. W F H h mgh J (b) the work done by gravity on the backpack. W -mgh J (gravitational force in opposite direction to motion) (c) the net work done on the backpack. For simplicity. W net W hiker + W gravity Copyright 009 Pearson Education, Inc. 6

7 8/04/0 Conceptual Example 7-3: Does the Earth do work on the Moon? The Moon revolves around the Earth in a nearly circular orbit, with approximately constant tangential speed, kept there by the gravitational force exerted by the Earth. Does gravity do (a) positive work, (b) negative work, (c) no work at all on the Moon? Since the only force is at right angles to the motion (cos90 0) no work is done by gravity on the moon Copyright 009 Pearson Education, Inc. 7- Scalar Product of Two Vectors Definition of the scalar, or dot, product: Therefore, we can write: Copyright 009 Pearson Education, Inc. 7

8 8/04/0 7- Scalar Product of Two Vectors Example 7-4: Using the dot product. The force shown has magnitude F P 0 N and makes an angle of 30 to the ground. Calculate the work done by this force, using the dot product, when the wagon is dragged 00 m along the ground. r r W d F d cos θ 0 00 cos J F P P Copyright 009 Pearson Education, Inc. 7-3 Work Done by a Varying Force Particle acted on by a varying force. Clearly, d is not constant! F r Copyright 009 Pearson Education, Inc. 8

9 8/04/0 7-3 Work Done by a Varying Force For a force that varies, the work can be approximated by dividing the distance up into small pieces, finding the work done during each, and adding them up. Copyright 009 Pearson Education, Inc. 7-3 Work Done by a Varying Force In the limit that the pieces become infinitesimally narrow, the work is the area under the curve: Or: Copyright 009 Pearson Education, Inc. 9

10 8/04/0 7-3 Work Done by a Varying Force Work done by a spring force: The force exerted by a spring is given by Hooke s: r F S r kx Note that the force exerted by the spring is always in the opposite direction to the displacement. Copyright 009 Pearson Education, Inc. 7-3 Work Done by a Varying Force Plot of F vs. x. Work done is equal to the shaded area. Copyright 009 Pearson Education, Inc. 0

11 8/04/0 7-3 Work Done by a Varying Force Example 7-5: Work done on a spring. (a) A person pulls on a spring, stretching it 3.0 cm, which requires a maximum force of 75 N. How much work does the person do? F 75 Spring Constant : F kx k 500 N/m x 0.03 ( ) Work : W kx.j (b) If, instead, the person compresses the spring 3.0 cm, how much work does the person do? Work is the same as in (a) Copyright 009 Pearson Education, Inc. 7-3 Work Done by a Varying Force Example 7-6: Force as a function of x. A robot arm that controls the position of a video camera in an automated surveillance system is manipulated by a motor that exerts a force on the arm. The force is given by where F 0.0 N, x m, and x is the position of the end of the arm. If the arm moves from x 0.00 m to x m, how much work did the motor do? Copyright 009 Pearson Education, Inc.

12 8/04/0 Copyright 009 Pearson Education, Inc. 7-3 Work Done by a Varying Force W F( x) dx F0 F0 x + 6x F0 ( x x ) 0 x x 3 x 3 x x + 3 x x + 8x 0 3 (0.050) (0.00).0 ( ) + 8(0.0070) 3 6 x x 0 dx F 0.0 N x m x 0.00 m x m J 7-4 Kinetic Energy and the Work-Energy Principle Energy was traditionally defined as the ability to do work. We now know that not all forces are able to do work; however, we are dealing in these chapters with mechanical energy, which does follow this definition. Copyright 009 Pearson Education, Inc.

13 8/04/0 7-4 Kinetic Energy and the Work-Energy Principle Consider a car (any object) accelerated from r r r v to v by a constant force F The acceleration is calculated from : v v v a d The work done : W v v m d d We define the quantity translational kinetic energy of net mv F net mv net d mad mv to be the v the object + ad K mv Copyright 009 Pearson Education, Inc. 7-4 Kinetic Energy and the Work-Energy Principle This means that the work done is equal to the change in the kinetic energy: If the net work is positive, the kinetic energy increases. If the net work is negative, the kinetic energy decreases. Copyright 009 Pearson Education, Inc. 3

14 8/04/0 7-4 Kinetic Energy and the Work-Energy Principle Because work and kinetic energy can be equated, they must have the same units: kinetic energy is measured in joules. Energy can be considered as the ability to do work: A moving hammer strikes a nail and comes to rest. The hammer exerts a force F on the nail; the nail exerts a force -F on the hammer (Newton s third law). The work done on the nail by the hammer is positive (W n Fd >0). The work done on the hammer by the nail is negative (W h -Fd) Copyright 009 Pearson Education, Inc. 7-4 Kinetic Energy and the Work-Energy Principle Example 7-7: Kinetic energy and work done on a baseball. A 45 g baseball is thrown so that it acquires a speed of 5 m/s. (a) What is its kinetic energy? K mv J (b) What was the net work done on the ball to make it reach this speed, if it started from rest? As net work is equal to the change in kinetic energy, the net work also equals 45 J Copyright 009 Pearson Education, Inc. 4

15 8/04/0 7-4 Kinetic Energy and the Work-Energy Principle Example 7-8: Work on a car, to increase its kinetic energy. How much net work is required to accelerate a 000 kg car from 0 m/s to 30 m/s? W net ( 0 ).5 0 J ( ) K K mv mv m v v Copyright 009 Pearson Education, Inc. Example 7-9: Work to stop a car. A car traveling 60 km/h can brake to a stop within a distance d of 0 m. If the car is going twice as fast, 0 km/h, what is its stopping distance? Assume the maximum braking force is approximately independent of speed. If the car s velocity is doubled then the kinetic energy is increased by a factor of 4 (K v ). If the force is constant the stopping distance will be m Copyright 009 Pearson Education, Inc. 5

16 8/04/0 Example 7-0: A compressed spring. A horizontal spring has spring constant k 360 N/m. (a) How much work is required to compress it from its uncompressed length (x 0) to x.0 cm? W kx ( 0.0) J Copyright 009 Pearson Education, Inc. Example 7-0: A compressed spring. A horizontal spring has spring constant k 360 N/m. Spring is compressed.0 cm 0.0 m (a) W.8 J (b) If a.85 kg block is placed against the spring and the spring is released, what will be the speed of the block when it separates from the spring at x 0? Ignore friction. W v K K m mv m/s Copyright 009 Pearson Education, Inc. 6

17 8/04/0 Example 7-0: A compressed spring. A horizontal spring has spring constant k 360 N/m. (a) (b) (c) F D W v How much work is required to compress it from its uncompressed length (x 0) to x.0 cm? (.8 J) If a.85 kg block is placed against the spring and the spring is released, what will be the speed of the block when it separates from the spring at x 0? Ignore friction. (.54 m/s) Repeat part (b) but assume that the block is moving on a table and that some kind of constant drag force F D 7.0 N is acting to slow it down, such as friction. does negative work D F K W W K m D Copyright 009 Pearson Education, Inc. D x J J.4.3 m/s.85 Summary of Chapter 7 Work: Work done by a variable force: Kinetic energy is energy of motion: Copyright 009 Pearson Education, Inc. 7

18 8/04/0 Summary of Chapter 7 Work-energy principle: The net work done on an object equals the change in its kinetic energy. Copyright 009 Pearson Education, Inc. 8

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