Chapter 7 Lecture Chapter 7: Work and Energy © 2016 Pearson Education, Inc.

Work and Energy 1.Work Energy  Work done by a constant force (scalar product)  Work done by a varying force (scalar product & integrals) 2.Kinetic Energy Work and Energy Work-Energy Theorem

Forms of Mechanical Energy Work and Energy

An Overview of Energy Energy is conserved. Kinetic energy describes motion and relates to the mass of the object and its speed squared. Energy on earth originates from the sun. Energy on earth is stored thermally and chemically. Chemical energy is released by metabolism. Energy is stored as potential energy in object height and mass and also through elastic deformation. © 2016 Pearson Education, Inc.

Work and Energy CONSERVATION OF ENERGY

Internal Energy Can be "Lost" as Heat Atoms and molecules of a solid can be thought of as particles vibration randomly on spring like bonds. This vibration is an an example of internal energy. Energy can be dissipated by heat (motion transferred at the molecular level). This is referred to as dissipation. © 2016 Pearson Education, Inc.

Work and Energy Work by a Baseball Pitcher A baseball pitcher is doing work on the ball as he exerts the force over a displacement. v 1 = 0 v 2 = 44 m/s

Consider Only Parallel F and S – Figure 7.9 Forces applied at angles must be resolved into components. W is a scalar quantity that can be positive, zero, or negative. If W > 0 (W < 0), energy is added to (taken from) the system. © 2016 Pearson Education, Inc.

Applications of Force and Resultant Work – Figure 7.10 © 2016 Pearson Education, Inc.

Work and Energy Work Done by a Constant Force (I) Work (W)  How effective is the force in moving a body ? W [Joule] = ( F cos  ) d  Both magnitude (F) and directions (  ) must be taken into account.

Work and Energy Work Done by a Constant Force (II) Example: Work done on the bag by the person..  Special case: W = 0 J a) W P = F P d cos ( 90 o ) b) W g = m g d cos ( 90 o )  Nothing to do with the motion

Work and Energy Example 1A A 50.0-kg crate is pulled 40.0 m by a constant force exerted (F P = 100 N and  = 37.0 o ) by a person. A friction force F f = 50.0 N is exerted to the crate. Determine the work done by each force acting on the crate.

Work and Energy Example 1A (cont’d) W P = F P d cos ( 37 o ) W f = F f d cos ( 180 o ) W g = m g d cos ( 90 o ) W N = F N d cos ( 90 o ) 180 o 90 o d F.B.D.

Work and Energy Example 1A (cont’d) W P = 3195 [J] W f = [J] (< 0) W g = 0 [J] W N = 0 [J] 180 o

Work and Energy Example 1A (cont’d) W net =  W i = 1195 [J] (> 0)  The body’s speed increases.

Work Done By Several Forces – Example 7.3 © 2016 Pearson Education, Inc.

Work and Energy Related – Example 7.4 Using work and energy to calculate speed. Returning to the tractor pulling a sled problem of Example 7.3: If you know the initial speed, and the total work done, you can determine the final speed after displacement s. © 2016 Pearson Education, Inc.

Work and Energy Work-Energy Theorem W net = F net d = ( m a ) d = m [ ( v 2 2 – v 1 2 ) / 2d ] d = (1/2) m v 2 2 – (1/2) m v 1 2 = K 2 – K 1

© 2016 Pearson Education, Inc. Kinetic energy

Work and Energy Example 2 A car traveling 60.0 km/h to can brake to a stop within a distance of 20.0 m. If the car is going twice as fast, 120 km/h, what is its stopping distance ? (a) (b)

Work and Energy Example 2 (cont’d) (1) W net = F d (a) cos 180 o = - F d (a) = 0 – m v (a) 2 / 2  - F x (20.0 m) = - m (16.7 m/s) 2 / 2 (2) W net = F d (b) cos 180 o = - F d (b) = 0 – m v (b) 2 / 2  - F x (? m) = - m (33.3 m/s) 2 / 2 (3) F & m are common. Thus, ? = 80.0 m

Work and Energy Does the Earth do work on the satellite? Satellite in a circular orbit

What can you say about the work the sun does on the earth during the earth's orbit? (Assume the orbit is perfectly circular.) a)The sun does net negative work on the earth. b)The sun does net positive work on the earth. c)The work done on the earth is positive for half the orbit and negative for the other half, summing to zero. d)The sun does no work on the earth at any point in the orbit. © 2016 Pearson Education, Inc. Clicker question

B Riding a loop-the-loop starting at rest from a point A, find the minimum height h, so that the car will not fall of the track at the top of the circular part of the loop, which has a radius of 20m.

Work and Energy ForcesForces Forces on a hammerhead

Work Done By a Varying Force In Section 7.2, we defined work done by a constant force. Work by a changing force is sometimes considered. On a graph of force as a function of position, the total work done by the force is represented by the area under the curve between the initial and final positions. © 2016 Pearson Education, Inc.

Work and Energy Spring Force (Hooke’s Law) F S (x) = - k x FPFP FSFS Natural Length x > 0 x < 0 Spring Force (Restoring Force): The spring exerts its force in the direction opposite the displacement.

Work and Energy Work Done to Stretch a Spring Area ( triangle) = baseline * height/ 2 F S (x) = - k x W Natural Length FPFP FSFS k x · x/2=A

Work Done By a Varying Force Hooke's law: As seen, this is a prime example of a varying force. The work done by a stretching/compressing a spring is equal to the area of the shaded triangle, or © 2016 Pearson Education, Inc.

Work and Energy Example 1A A person pulls on the spring, stretching it 3.0 cm, which requires a maximum force of 75 N. How much work does the person do ? If, instead, the person compresses the spring 3.0 cm, how much work does the person do ?

Work and Energy (a) Find the spring constant k k = F max / x max = (75 N) / (0.030 m) = 2.5 x 10 3 N/m (b) Then, the work done by the person is W P = (1/2) k x max 2 = 1.1 J Example 1A (cont’d)

The two cylinders in the figure have masses m A > m B. The larger cylinder is attached to a spring that is stretched a distance s from its equilibrium length. A latch keeps the spring stretched and the system stationary. What is the system's total mechanical energy? a)m A gh A + m B gh B – 1/2kh A 2 b)m A gh A – m B gh B + 1/2kh A 2 c)m A gh A + m B gh B – 1/2ks 2 d)m A gh A + m B gh B + 1/2ks 2 © 2016 Pearson Education, Inc. Clicker question

Work and Energy Example 2 A 1.50-kg block is pushed against a spring (k = 250 N/m), compressing it m, and released. What will be the speed of the block when it separates from the spring at x = 0? Assume  k = (i) F.B.D. first ! (ii) x < 0 F S = - k x

Work and Energy (a) The work done by the spring is (b) W f = -  k  F N (x 2 – x 1 ) = ( ) (c) W net = W S + W f = x (d) Work-Energy Theorem: W net = K 2 – K 1  4.12 = (1/2) m v 2 – 0  v = 2.34 m/s W S =1/2 k x 2 = ½ (250N/m) ( 0.2 m 2 )= J Example 2 (cont’d)

Energy Conservation 1.Conservative/Nonconservative Forces  Work along a path (Path integral)  Work around any closed path (Path integral) 2.Potential Energy Potential Energy and Energy Conservation Mechanical Energy Conservation

© 2016 Pearson Education, Inc. Gravitational potential energy

Energy Conservation Work Done by the Gravitational Force (III) W g y 1 W g > 0 if y 2 < y 1 The work done by the gravitational force depends only on the initial and final positions..

Energy Conservation Work Done by the Gravitational Force (IV) W g(A  B  C  A) = W g(A  B) + W g(B  C) + W g(C  A) = mg(y 1 – y 2 ) mg(y 2 - y 1 ) = 0 dldl A B C

Energy Conservation Work Done by the Gravitational Force (V) W g = 0 for a closed path The gravitational force is a conservative force.

Energy Conservation Work Done by F f (I) (Path integral) - μmg L LBLB LALA L depends on the path. Path A Path B

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Energy Conservation The work done by the friction force depends on the path length. The friction force: (a) is a non-conservative force; (b) decreases mechanical energy of the system. W f = 0 (any closed path) Work Done by F f (II)

Energy Conservation Example 1 A 1000-kg roller-coaster car moves from point A, to point B and then to point C. What is its gravitational potential energy at B and C relative to point A?

Energy Conservation W g(A  C) = U g (y A ) – U g (y C ) W g(A  B  C) = W g(A  B) + W g(B  C) = mg(y A - y B ) + mg(y B - y C ) = mg(y A - y C ) dldl A B C y A B

Work and Energy Conservation of Energy (Sections 7.6 and 7.7) When only conservative forces act on an object, the total mechanical energy (kinetic plus potential) is constant; that is,Ki + Ui = Kf + Uf, where U may include both gravitational and elastic potential energies. If some of the forces are non conservative, we label their work as Wother. The change in total energy (kinetic plus potential)of an object, during any motion, is equal to the work Wother done by the non conservative forces: Ki + Ui + Wother = Kf + Uf (Equation 7.17). Non conservative forces include friction forces, which usually act to decrease the total mechanical energy of a system.

Climbing the Sear tower Work and Energy

Power

The Burj Khalifa is the largest man made structure in the world and was designed by Adrian Smith class of 1966 thebatt.com Febuary 25 th