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Transcript
Lecture 12 Goals Analyze situations with Work and a Constant Force Analyze situations with Work and a Varying Force Introduce Hooke’s Law springs Relate external or “net” work to speed Introduce concepts of Kinetic and Potential energy Physics 201: Lecture 12, Pg 1 Definition of Work, The basics Ingredients: Force ( F ), displacement ( r ) Work, W, of a constant force F acts through a displacement r : W ≡ F · r F r (Work is a scalar) Work tells you something about what happened on the path! Did something do work on you? Did you do work on something? If only one force acting: Did your speed change? Physics 201: Lecture 12, Pg 2 Net Work: 1-D Example (constant force) A force F = 10 N pushes a box across a frictionless floor for a distance x = 5 m. Finish Start F = 0° x Net Work is F x = 10 x 5 N m = 50 J 1 Nm ≡ 1 Joule and this is a unit of energy Physics 201: Lecture 12, Pg 3 Net Work: 1-D 2nd Example (constant force) A force F = 10 N is opposite the motion of a box across a frictionless floor for a distance x = 5 m. Finish Start F = 180° x Net Work is F x = -10 x 5 N m = -50 J Physics 201: Lecture 12, Pg 4 Work: “2-D” Example (constant force) An angled force, F = 10 N, pushes a box across a frictionless floor for a distance x = 5 m and y = 0 m Start F Finish = -45° Fx x (Net) Work is Fx x = F cos(-45°) x = 50 x 0.71 Nm = 35 J Physics 201: Lecture 12, Pg 5 Infinitesimal Work vs speed along a linear path Let dW Fx dx F Start F Finish Fx ma x dx m dvx dt m dv x dx dx x dt m dvx vx Physics 201: Lecture 12, Pg 6 Work vs speed along a linear path dW Fx dx m vx dvx xf vxf xi If F is constant vxi Wext Fx dx mvx dvx xf vxf xi vxi Wext Fx dx m vx dvx Fx ( x f xi ) Fx x 1 2 2 mv xf 1 2 2 mv xi K K is defined to be the change in the kinetic energy Physics 201: Lecture 12, Pg 7 Work in 3D…. x, y and z with constant F: Fx ( x f xi ) Fx x Fy ( y f yi ) Fy y 1 2 mv xf 2 1 2 mv yf 2 1 2 mv xi 2 1 2 mv yi 2 1 2 mv zi 2 1 2 Fz ( z f zi ) Fz z 2 mv zf 1 2 1 2 Fx x Fy y Fz z 2 mv f 2 mvi with v 2 2 vx 2 vy 2 vz K v v Kinetic energy : KE mv 1 2 Physics 201: Lecture 12, Pg 8 2 Examples of External Work K = Wext Pushing a box on a smooth floor with a constant force; there is an increase in the kinetic energy Examples of No External or Net Work Pushing a box on a rough floor at constant speed Driving at constant speed in a horizontal circle Holding a book at constant height This last statement reflects what we call the “system” ( Dropping a book is more complicated because it “potentially” involves changes in the “potential” energy. The answer depends on what we call the system. ) Physics 201: Lecture 12, Pg 10 Exercise Work in the presence of friction and non-contact forces A box is pulled up a rough (m > 0) incline by a rope-pulleyweight 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 v A. B. C. D. 2 3 4 5 Physics 201: Lecture 12, Pg 11 Work and Varying Forces (1D) Area = Fx x F is increasing Here W = F · r becomes dW = F dx Consider a varying force F(x) Fx xf x x Wext F ( x ) dx Finish Start F F xi = 0° x Work has units of energy and is a scalar! Physics 201: Lecture 12, Pg 12 A Hooke’s Law spring Most springs have a force which increases linearly with displacement from the equilibrium F ( x xeq ) k ( x xeq ) spring at an equilibrium position 50 m Force (N) 40 x 30 20 F spring compressed 10 1.0 x 2.0 3.0 4.0 Displacement (m) F Slope: spring constant k Physics 201: Lecture 12, Pg 13 • Example: Work Kinetic-Energy Theorem with variable force How much will the spring compress (i.e. x = xf - 0) to bring the box to a stop (i.e., v = 0 ) if the object is moving initially at a constant velocity (vo) on frictionless surface as shown below ? Wext f F r dr i x x f Fx dx x x i t o vo eq xf eq Wext m kx dx 0 spring at an equilibrium position Wext x - 2 kx 1 2 xf 0 - 2 k x K 2 f 1 V=0 t | F - k x m0 mv m 1 2 spring compressed 2 1 2 2 1 2 Physics 201: Lecture 12, Pg 14 2 0 Example: Work Kinetic-Energy Theorem with variable force • How much will the spring compress (i.e. x) to bring the box to a stop (i.e., v = 0 ) if the object is moving initially at a constant velocity (vo) on frictionless surface as shown below ? to vo Notice that the spring force is opposite the displacement m spring at an equilibrium position x For the mass m, work is negative, Ws V=0 t F m For the spring, spring compressed work is positive, Wapp Physics 201: Lecture 12, Pg 15 What happens when I release the spring? • How fast will the block be moving when it loses contact with the spring? 1 2 1 2 1 2 f 0 2 2 2 W k x mv mv V=0 F m spring compressed vo m spring now at an equilibrium position Physics 201: Lecture 12, Pg 16 Work done by friction A small block slides inside a hoop of radius r. The walls of the hoop are frictionless but the horizontal floor has a coefficient of sliding friction of m. The block’s initial velocity is v and is entirely tangential. How far, in angle around the circle does the block travel before coming to rest ? SFy = 0 = -mg + N SFx = m ar = m v2 /R Ff = m N = m mg W = Ff s = K = ½ mvf2 – ½ mvi2 = – ½ mv2 - m mg s = – ½ mv2 s = v2 / 2mg = r = v2 / (2mgr) Finish v Start R Physics 201: Lecture 12, Pg 17 Friction….again Looking down on an air-hockey table with no air flowing (m > 0). WPath 1 friction = Ff xpath 1 WPath 2 friction = Ff xpath 2 xpath 2 > xpath 1 so W2 > W1 Path 2 Path 1 Physics 201: Lecture 12, Pg 18 The spring and friction have an important difference For the spring I can reverse the process and recover the kinetic energy The compressed spring has the ability to do work For the spring the work done is independent of path The spring is said to be a conservative force In the case of friction there is no immediate way to back transfer the energy of motion In this case the work done is dependent on path Friction is said to be a non-conservative force Physics 201: Lecture 12, Pg 19 Potential Energy (U) For the compressed spring the energy is “hidden” but still has the ability to do work (i.e., allow for energy transfer) This kind of “energy” is called “Potential Energy” U ( x) spring k ( x xeq ) 1 2 2 The gravitation force, if constant, has the same properties. U ( y ) gravity mg ( y y0 ) Physics 201: Lecture 12, Pg 20 Mechanical Energy If only “conservative” forces, then total mechanical energy (potential U plus kinetic K energy) of a system is conserved For an object in a gravitational “field” ½ m vyi2 + mgyi = ½ m vyf2 + mgyf = constant K ≡ ½ mv2 U ≡ mgy Emech = K + U Emech = K + U = constant K and U may change, but Emech = K + U remains a fixed value. Emech is called “mechanical energy” Physics 201: Lecture 12, Pg 21 For Tuesday Read Chapter 8 Start HW6 Physics 201: Lecture 12, Pg 22