Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Hunting oscillation wikipedia , lookup
Classical central-force problem wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Relativistic mechanics wikipedia , lookup
Gibbs free energy wikipedia , lookup
Heat transfer physics wikipedia , lookup
Eigenstate thermalization hypothesis wikipedia , lookup
General Physics I Lecture 5: Conservation of Energy Prof. WAN, Xin (万歆) [email protected] http://zimp.zju.edu.cn/~xinwan/ Outline ● Revisit work-kinetic energy theorem W =K f −K i=Δ K ● ● Introduce potential energy (for conservative forces) W =U i−U f =−Δ U Conservation of (mechanical) energy E=K +U=const ● Newton's second law Non-conservative forces Work-Kinetic Energy Theorem ● ● ● The net work done on a particle by the net force acting on it is equal to the change in the kinetic energy of the particle. We can think of the kinetic energy as the work a particle can do in coming to rest, or the amount of energy stored in the particle. What about other forms of energy? Constant Acceleration Set ti 0, t f t , v xf v xi a x t 1 x f xi v xi v xf t 2 1 2 v xi t a x t 2 2 v xf v xi 2a x 2 v xf vx axt 2 a xt 2 v xi 0 t Time to Forget ● Work done W =max (x f −xi ) ● ● Kinetic energy change 1 1 2 2 K f −K i= m v xf − mv xi 2 2 From the work-kinetic energy theorem v 2xf −v2xi x f −x i= 2 ax Potential Energy ● ● ● Roughly speaking, the energy associated with a system of objects. Consider a system of two particle-like objects. The work done by the force acting on one of the objects causes a transformation of energy between the object's kinetic energy and other forms of the system's energy. Concrete examples: − Elastic potential energy − Gravitational potential energy Elastic Potential Energy ● The elastic potential energy of the system can be thought of as the energy stored in the deformed spring (one that is either compressed or stretched from its equilibrium position). xf 1 1 2 2 W s = ∫ (−kx) dx = k x i − k x f 2 2 x i = U i−U f = −(U f −U i ) = −Δ U g Elastic Potential Energy ● The elastic potential energy associated with the system is defined by − Zero: undeformed − Always positive in a deformed spring Gravitational Potential Energy ● Gravitational potential energy is the potential energy of the object–Earth system. The potential energy is transformed into kinetic energy of the system by the gravitational force. − The earth is much more massive than the object. The earth can be modeled as stationary, and the kinetic energy of the system can be represented entirely by the kinetic energy of the lighter object. − The kinetic energy of the system is represented by that of the object falling toward the Earth. Gravitational Potential Energy ● The work done by the gravitational force ⃗ ⃗ )⋅d W g=(m g =(−m g ŷ )⋅( y f − y i ) ŷ =m g y i −m g y f =U i−U f =−(U f −U i )=−Δ U g Gravitational Potential Energy ● If a particle of mass m is at a distance y above the Earth’s surface, the gravitational potential energy of the particleEarth system is Independent of the choice of the origin. Valid only for objects near the surface of the Earth. Frictional Force Although the details of friction are quite complex at the atomic level, this force ultimately involves an electrical interaction between atoms or molecules. Force of static friction Force of kinetic friction Frictional Force ● Experimentally, we find that, to a good approximation, both fs,max and fk are proportional to the normal force acting on the object. μ s: coefficient of static friction μ k: coefficient of dynamic friction Slow Down, Please! A 1 500-kg car moving on a flat, horizontal road which curves with a radius 35.0 m and the coefficient of static friction is 0.500, find the maximum speed the car can make the turn successfully. Work Done by Frictional Force ⃗ W friction= f⃗k⋅d=− f kd Can we define a frictional potential like we did in the case of gravitational force? Example ● Jumping from 2.00 m high 1 2 mgh= m v f −0 2 v f =√2 g h=6.26 m/ s ● Sliding down 1 2 mgh− f̄k l= m v f −0 2 Depending on path! Conserved or Nonconserved? ● The path makes no difference when we consider the work done by the gravitational force, but it does make a difference when we consider the energy loss due to frictional forces. − Gravitational force is conservative. − Frictional force is nonconservative. Conservative Forces ● ● A force is conservative if the work it does on a particle moving between any two points is independent of the path taken by the particle. The work done by a conservative force on a particle moving through any closed path is zero. (A closed path is one in which the beginning and end points are identical.) A B Conservation of Mechanical Energy ● The work done by a conservative force F to a particle is ● On the other hand, we have ● Together, we have Conservation of Mechanical Energy ● The total mechanical energy of a system remains constant in any isolated system of objects that interact only through conservative forces. − No energy is added to or removed from the system. − No nonconservative forces are doing work within the system. − Sum over the potential energy associated with each conservative force. Nonconservative Forces ● ● ● Frictional force Δ E=Δ K +Δ U =W friction By the frictional force, mechanical energy is transformed into internal energy (the kinetic energy associated with the random motions of the atoms or molecules and the potential energy associated with the forces between the atoms or molecules). We will revisit the issue during the discussion of the first law of thermodynamics. Energy is conserved after taking into account the internal energy. Walter Lewin Landing of the Actor The largest tension determines the maximum angle without lifting the sandbag. Conservative Force ● ● Earlier, we learned how to determine the change in potential energy of a system when we are given the conservative force. Now we show how to find the force if the potential energy of the system is known. Derivative of Potential Energy ● ● Any conservative force acting on an object within a system equals the negative derivative of the potential energy of the system with respect to x. Deformed spring dU s d 1 2 F s =− =− k x =−kx dx dx 2 ( ● ) Gravitation dU g d(mgy ) F g=− =− =−mg dy dy Vector Form ● In the language of vector calculus, F equals the negative of the gradient of the scalar quantity U(x, y, z). In three dimensions, You will learn in the case of electric field and electric potential Equilibrium of a System In general, positions of stable equilibrium correspond to points for which U(x) is a minimum. dU =0 dx 2 d U >0 2 dx Unstable Equilibrium In general, positions of unstable equilibrium correspond to points for which U(x) is a maximum. dU =0 dx 2 d U <0 2 dx Lennard–Jones Potential ● The potential energy associated with the force between two neutral atoms in a molecule can be modeled by the Lennard–Jones potential energy function: Mass-Energy Equivalence ● ● ● Energy can never be created or destroyed. Energy may be transformed from one form to another, but the total energy of an isolated system is always constant. Conservation of mass states that in any physical or chemical process, mass is neither created nor destroyed. We will learn in the theory of special relativity mass and energy are not separately conserved. Quantization of Energy