Phys205A_Lecture17

Physics for Scientists & Engineers with Modern Physics (4th Edition)

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Chapter 7 Kinetic and Potential Energy of a System Lecture 17-18
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Work Done by a Varying Force z Assume that during a very small displacement, x , F is constant z For that displacement, W ~ F x z For all of the intervals, ≈∆ f i x x x WF x X F F x W = F D x cos a What if F changes the value and/or direction?
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Work Done by a Varying Force, cont z z Therefore, z The work done is equal to the area under the curve between x i and x f ∆→ ∆= lim 0 f f i i x x xx x x x Fx F d x = f i x x x WF d x Eample:
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Work Done By Multiple Forces z If more than one force acts on a system and the system can be modeled as a particle , the total work done on the system is the work done by the net force () == ∑∑ f i x net x x WW F d x
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Work Done by Multiple Forces, cont. z In the general case of a net force whose magnitude and direction may vary z If the system cannot be modeled as a particle, then the total work is equal to the algebraic sum of the work done by the individual forces z Remember work is a scalar, so this is the algebraic sum = net by individual forces WW () == ∑∑ Fr r r f i x net x d m1 m2 m3 L1 L2 W-?
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Example of varying force: Work Done By A Spring z A model of a common physical system for which the force varies with position z The block is on a horizontal, frictionless surface z The force exerted by the spring is always directed opposite to the displacement from equilibrium z The spring force is sometimes called the restoring force z If the block is released it will oscillate back and forth between – x and x
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Hooke’s Law z The force exerted by the spring is F s = - kx z x is the position of the block with respect to the equilibrium position ( x = 0) z k is called the spring constant or force constant and measures the stiffness of the spring z This is called Hooke’s Law
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Hooke’s Law, cont. z When x is positive (spring is stretched), F is negative z When x is 0 (at the equilibrium position), F is 0 z When x is negative (spring is compressed), F is positive
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Work Done by a Spring z Identify the block as the system z Calculate the work as the block moves from x i = - x max to x f = 0 z Interestingly: The total work done as the block moves from x max to x max is zero () == = ∫∫ max 0 2 1 2 f i x sx xx W F dx kx dx kx
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Work Done by a Spring, cont. z Assume the block undergoes an arbitrary displacement from x = x i to x = x f z The work done by the spring on the block is z If the motion ends where it begins, W = 0 () =− = 22 11 f i x si f x Wk x d x k x k x
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Spring with an Applied Force z Suppose an external agent, F app , stretches the spring z The applied force is equal and opposite to the spring
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Phys205A_Lecture17 - Lecture 17-18 Chapter 7 Kinetic and...

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