Phys04 - Unit 4 Work and Energy 4.1 4.2 4.3 4.4 4.5 4.6...

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1 Unit 4 Work and Energy 4.1 Work and kinetic energy 4.2 Work - energy theorem 4.3 Potential energy 4.4 Total energy 4.5 Energy diagrams 4.6 A general study of the potential energy curve 4.1 Work and kinetic energy (1) Work: unit in joule (J) Work done by the force F d F = = = d F d F W x ) cos ( θ F x : the force along the displacement Remark: Positive work is done on an object when the point of application of the force moves in the direction of the force. Mathematical statement: Work done is the dot product of the force and the displacement. Dot product: AB ⋅= = + + AB A B A B A B xx yy zz cos Example A 75.0-kg person slides a distance of 5.00m on a straight water slide, dropping through a vertical height of 2.50 m. How much work does gravity do on the person? y x F G F G d G d =5.00 m h= 2.5 mg
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2 Answer: The gravity along the displacement has a magnitude mg cos θ . By the definition of work done, the work done of the gravity on the person is given by W = ( mg cos ) d = mg ( h / d ) d = mgh = (75.0)(9.8)(2.50) = 1837.5 J Remark: When F or d is not a constant (see the left figure, the movement of the particle is always varying). i i w s F i = W ii i =⋅ Fs where s i is extremely small. The work done of the gravity on the ball’s falling down is given by mgh y mg y F y x F - d W h h C C = = = + = = 0 0 d d ) ˆ d ˆ d ( ˆ j i j s F (2) Power: work done per unit time Power == = lim t W t dW dt d dt 0 Since dt d s v = , we have the power equals to the dot product of force and velocity. e.g. Power v F = If the force and the velocity are in the same direction, P = Fv. (3) Kinetic energy 2 2 1 mv K 4.2 Work - energy theorem If an object is moving with an acceleration a and a distance d is moved, then according to the formula ad v v i f 2 2 2 + = , we have d m F v v i f ) ( 2 2 2 + = . The work done on the object W is given by i f i f i f K K mv mv v v m Fd W = = = = 2 2 2 2 2 1 2 1 ) ( 2 Or, we can rewrite it as fi WK K K =− = . y x mg h C ( h , h ) (0, 0) F G F G d G v i v f
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3 The work-energy theorem states that the total work done on the object by the forces equals the change in kinetic energy. Remark: A proof by using integration. Suppose a particle moves in a straight line (1-D) WF x d x x x i f = () Fm am dv dt == , v dx dt = ∴= = = ∫∫ Wm dv dt dx mdv dx dt mvdv x x x x v v i f i f i f () =−= 1 2 2 1 2 2 mv mv K K fi f i 4.3 Potential energy The gravitational potential energy U = mgh The difference of the potential energy: U = U f U i = W , where W is the work done by gravity if the block is released. In this case, W is positive. When a spring is compressed by an amount x , the work done of the applied force 2 0 0 2 1 kx kxdx Fdx x x = = = ( F : applied force) Uk x = 1 2 2 potential energy of a spring.
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This note was uploaded on 09/06/2010 for the course BSC PHY1417 taught by Professor Prof during the Spring '08 term at HKU.

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Phys04 - Unit 4 Work and Energy 4.1 4.2 4.3 4.4 4.5 4.6...

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