Lecture 15More Work: “2-D” Example (constant force)Work in 3D….(assigning U to be external to the system)A tool: Scalar Product (or Dot Product)Scalar Product (or Dot Product)Definition of Work, The basicsExercise Work in the presence of friction and non-contact forcesExercise Work in the presence of friction and non-contact forcesWork and Varying Forces (1D)Slide 13Slide 15Conservative Forces & Potential EnergyConservative Forces and Potential EnergyExercise Work Done by GravityNon-conservative Forces :A Non-Conservative Force, FrictionA Non-Conservative ForceWork & Power:Slide 27Slide 28Exercise Work & PowerChap. 12: Rotational DynamicsSlide 31Rotational Dynamics: A child’s toy, a physics playground or a student’s nightmareOverview (with comparison to 1-D kinematics)Slide 37Physics 207: Lecture 15, Pg 1Lecture 15Goals:Goals:•Chapter 11Chapter 11 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 rotation about a fixed axis. Employ “conservation of angular momentum” conceptAssignment: HW7 due March 25th 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 2More Work: “2-D” Example (constant force)(Net) Work is Fx x = F cos(= F cos(-45°) x = 50 x 0.71 Nm = 35 J = 50 x 0.71 Nm = 35 J Notice that work reflects energy transferxFAn angled force, F = 10 N, pushes a box across a frictionless floor for a distance x = 5 m and y = 0 m = -45° StartFinishFxPhysics 207: Lecture 15, Pg 4Work in 3D….(assigning U to be external to the system)221221)(zizfzifzmvmvzFzzF x, y and z with constant F:221221)(yiyfyifymvmvyFyyF 221221)(xixfxifxmvmvxFxxF 2222221221 with zyxifzyxvvvvKmvmvzFyFxFPhysics 207: Lecture 15, Pg 5Useful for performing projections.A - î = Axî - î = 1 î - j = 0îAAxAyA tool: Scalar Product (or Dot Product)Calculation also in terms of magnitudes and relative angles.A - B = (Ax )(Bx) + (Ay )(By ) + (Az )(Bz )Calculation can be made in terms of components.A - B ≡ | A | | B | cos You choose the way that works best for you!A · B ≡ |A| |B| cos()Physics 207: Lecture 15, Pg 6Scalar Product (or Dot Product)Compare:A - B = (Ax )(Bx) + (Ay )(By ) + (Az )(Bz )Redefine A F (force), B r (displacement) Notice:F - r = (Fx )(x) + (Fy )(z ) + (Fz )(z)So hereF - r = WMore generally a Force acting over a Distance does WorkPhysics 207: Lecture 15, Pg 7Definition of Work, The basicsIngredients: Force ( F ), displacement ( r )“Scalar or Dot Product” rdisplacementFWork, W, of a constant force F acts through a displacement r :W = F · r (Work is a scalar)(Work is a scalar)If we know the angle the force makes with the path, the dot product gives us F cos and rIf the path is curved at each pointand rdFdWrdFfirrrdFWPhysics 207: Lecture 15, Pg 10ExerciseWork 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 11Exercise Work in the presence of friction and non-contact forcesA box is pulled up a rough ( > 0) incline by a rope-pulley-weight arrangement as shown below. How many forces are doing work on the box ? And which are positive (T) and which are negative (f, mg)?(For mg only the component along the surface is relevant) Use a Free Body Diagram (A) 2 (B) 3 is correct (C) 4 (D) 5vfmgNTPhysics 207: Lecture 15, Pg 12Work and Varying Forces (1D)Consider a varying force F(x)FxxxArea = Fx xF is increasingHere W = F · r becomes dW = F dx F = 0° StartFinishWork has units of energy and is a scalar!fixxdxxFW )(FxPhysics 207: Lecture 15, Pg 13•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 and xi is the equilibrium position of the spring?xvimtiFspring compressedspring at an equilibrium positionV=0tmfixxdxxFW )(boxfiixxdxxxkW )(-boxfiixxxxkW|221box)( - K 0 )( -221221box kxxkifW2i212 21221v0 - mmxk Example: Hooke’s Law Spring (xi equilibrium)Physics 207: Lecture 15, Pg 15Work 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 16Conservative 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.This can be written as:W = F ·dr = - U U = Uf - Ui = - W = - F • drrirfrirf Uf UiPhysics 207: Lecture 15, Pg 17Conservative 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 18 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 23Non-conservative Forces :If the work done does not depend on the path taken, the force involved is said to be
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