Lecture 15Scalar Product (or Dot Product)Work in terms of the dot productEnergy and WorkSlide 7Exercise Work in the presence of friction and non-contact forcesWork and Varying Forces (1D)Slide 10Slide 11Conservative Forces & Potential EnergyConservative Forces and Potential EnergyExercise Work Done by GravityA Non-Conservative ForceWork & Power:Slide 21Slide 22Exercise Work & PowerChap. 12: Rotational DynamicsRotational VariablesRotational Variables...Slide 28Physics 207: Lecture 15, Pg 1Lecture 15Goals:Goals:•Chapter 11Chapter 11 Employ the dot product Employ conservative and non-conservative forces Use the concept of power (i.e., energy per time)•Chapter 12Chapter 12 Extend the particle model to rigid-bodies Understand the equilibrium of an extended object. Understand rigid object rotation about a fixed axis. Employ “conservation of angular momentum” conceptAssignment: HW7 due March 10th For Thursday: Read Chapter 12, Sections 7-11do not concern yourself with the integration process in regards to “center of mass” or “moment of inertia””Physics 207: Lecture 15, Pg 2Useful for finding parallel componentsA - î = Axî - î = 1 î - ĵ = 0A - B = (Ax )(Bx) + (Ay )(By ) + (Az )(Bz )Calculation can be made in terms of components.Calculation also in terms of magnitudes and relative angles.Scalar Product (or Dot Product)A - B ≡ | A | | B | cos You choose the way that works best for you!cosBABA-îAAxAyĵPhysics 207: Lecture 15, Pg 4Work in terms of the dot productIngredients: Force ( F ), displacement ( r )Looks just like a Dot Product! rdisplacementFWork, W, of a constant force F acts through a displacement r :If the path is curved at each pointand rdFdWrdFfirrrdFWrFrFW-cosPhysics 207: Lecture 15, Pg 6Energy and WorkWork, W, is the process of energy transfer in which a force component parallel to the path acts over a distance; individually it effects a change in energy of the “system”.1. K or Kinetic Energy 2. U or Potential Energy (Conservative)and if there are losses (e.g., friction, non-conservative)3. ETh Thermal EnergyPositive W if energy transferred to a systemPhysics 207: Lecture 15, Pg 7A. U K B. U EThC. K UD. K EThE. There is no transformation because energy is conserved.A child slides down a playground slide at constant speed. The energy transformation isPhysics 207: Lecture 15, Pg 8ExerciseWork in the presence of friction and non-contact forcesA. 2B. 3C. 4D. 5A box is pulled up a rough ( > 0) incline by a rope-pulley-weight arrangement as shown below. How many forces (including non-contact ones) are doing work on the box ? Of these which are positive and which are negative? State the system (here, just the box) Use a Free Body Diagram Compare force and path vPhysics 207: Lecture 15, Pg 9Work and Varying Forces (1D)Consider a varying force F(x)FxxxArea = Fx xF is increasingHere W = F · r becomes dW = Fx dx F = 0° StartFinishWork has units of energy and is a scalar!fixxxdxxFW )(FxPhysics 207: Lecture 15, Pg 10•How much will the spring compress (i.e. x = xf - xi) to bring the box to a stop (i.e., v = 0 ) if the object is moving initially at a constant velocity (vi) on frictionless surface as shown below with xi = xeq , the equilibrium position of the spring?xvimtiFspring compressedspring at an equilibrium positionV=0tmfixxxdxxFW )(boxfixxeqdxxxkW )(-boxfiixxxxkW|221box)( - K 0 )( -221221box kxxkifW2i212 21221v0 - mmxk Example: Hooke’s Law Spring (xi equilibrium)Physics 207: Lecture 15, Pg 11Work signsxvimtiFspring compressedspring at an equilibrium positionV=0tmNotice that the spring force is opposite the displacementFor the mass m, work is negativeFor the spring, work is positive They are opposite, and equal (spring is conservative)Physics 207: Lecture 15, Pg 12Conservative Forces & Potential EnergyFor any conservative force F we can define a potential energy function U in the following way:The work done by a conservative force is equal and opposite to the change in the potential energy function.W = F ·dr ≡ - U rirf Uf UiPhysics 207: Lecture 15, Pg 13Conservative Forces and Potential EnergySo we can also describe work and changes in potential energy (for conservative forces)U = - WRecalling (if 1D)W = Fx xCombining these two,U = - Fx xLetting small quantities go to infinitesimals,dU = - Fx dxOr,Fx = -dU / dxPhysics 207: Lecture 15, Pg 14 ExerciseWork Done by GravityAn frictionless track is at an angle of 30° with respect to the horizontal. A cart (mass 1 kg) is released from rest. It slides 1 meter downwards along the track bounces and then slides upwards to its original position. How much total work is done by gravity on the cart when it reaches its original position? (g = 10 m/s2)1 meter30°(A) 5 J (B) 10 J (C) 20 J (D) 0 J h = 1 m sin 30° = 0.5 mPhysics 207: Lecture 15, Pg 19A Non-Conservative ForcePath 2Path 1Since path2 distance >path1 distance the puck will be traveling slower at the end of path 2. Work done by a non-conservative force irreversibly removes energy out of the “system”. Here WNC = Efinal - Einitial < 0 and reflects EthermalPhysics 207: Lecture 15, Pg 20Work & Power:Two cars go up a hill, a Corvette and a ordinary Chevy Malibu. Both cars have the same mass. Assuming identical friction, both engines do the same amount of work to get up the hill.Are the cars essentially the same ?NO. The Corvette can get up the hill quickerIt has a more powerful engine.Physics 207: Lecture 15, Pg 21Work & Power:Power is the rate at which work is done.InstantaneousPower:AveragePower:A person, mass 80.0 kg, runs up 2 floors (8.0 m). If they climb it in 5.0 sec, what is the average power used?Pavg = F h / t = mgh / t = 80.0 x 9.80 x 8.0 / 5.0 WP = 1250 W Example:Units (SI) areWatts (W):1 W = 1 J / 1stWPdtdWP Physics 207: Lecture 15, Pg 22Work & Power:Power is also,If force constant, W= F x = F ( v0 t + ½ at2 )and P = W / t = F (v0 + at) xxxvFPtxFtWP Physics 207: Lecture 15, Pg 23Exercise Work & PowerA. TopB. MiddleC. BottomStarting from rest, a car drives up a hill at constant acceleration and then quickly stops at the
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