chapter 3

Slide1: 

Chapter 13 the law of conservation of energy

Slide2: 

13.1 toward an idea of energy kinetic energy potential energy Energy is conserved

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13.2 work and potential energy 13.2.1 work 3. work done by a variable force that is not in the direction of the displacement 2. work done by a variable force 1. work done by a constant force

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1）work is a scalar. Positive or negative work is determined by the angleα 　　 0≤α ＜900 positive α=900 F doesn’t do work 900＜α≤1800 negative Note: 3）work is related to the path a moving body follows.

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[example 1] What amount of work is done by a spring force exerted by a spring that is stretched from one position A to another B ? solution：when we stretch a spring a distance x, the spring exerts a force By the definition of work,when the body is moved from A to B, the work the spring force does is stretched，negative work；compressed, positive work.

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[example 2] What amount of work is done by gravity when a body moves from M to M0 ? solution:

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[example 3] What amount of work is done by gravitational force when a body of mass m revolves around another body of mass M from a to b along its elliptical orbit ? solution：

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average instantaneous 4. power Example A block of mass m moves on a smooth inclined plane fixed on a table. When the block slides at a vertical height h , find the instantaneous power of gravity acting on the block.

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1. conservative and non-conservative forces Work done by a conservative force following a closed path is zero. proof： 13.2.2 potential energy

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2、potential energy The energy associated with a body’s location is stored in a conservative field. The potential energy of a body at a certain position M is equal to the work done by a conservative force acting on the body moving from this position to the ground position M0 . 3、potential energy in some typical conservative fields 1）gravity

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2）gravitation The potential energy at a some point is given by Here, we choose ground point far away from the given point. Equilibrium is the ground point. 3）spring force

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4、the relationship between potential energy and work done by a conservative force The work done by a conservative force is equal to the change in potential energy.

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[example 1] A cloud stone at a height h above the surface of the earth falls downward. If we neglect air resistance, what amount of work does gravitational force do on the cloud stone in the process of its fall? solution： Let the center of the earth to be the origin. Gravitational force and position vector have opposite direction. another

Slide14: 

13.3 the law of conservation of energy 13.3.1 kinetic energy 1、definition 2、use momentum to describe kinetic energy

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13.3.2 the theorem of kinetic energy The work done by net force on a particle is equal to the change in kinetic energy of the particle. 1. for a particle

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[example 1] A block of mass m and speed v0 enters a hemicycle barrier fixed on a horizontally smooth table,as the figure shows. Prove that when the block moves off the other end of the barrier, work done by friction exerted by the barrier on the block is where μ is the frictional coefficient. proof： from Newton’s second law and we have That is,

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By integration, So that the speed at which the block moves off the barrier is obtained According to the theorem of kinetic energy, the work done by the friction is given by we get

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2、for a system external forces: internal forces: initial velocities: final velocities: The paths these two particles follows is shown in the figure. Applying the theorem of kinetic energy to these two particles respectively, we get to m1: to m2:

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Adding equation (1) to equation (2), we obtain Wexternal Winternal So it becomes The work done by external and internal forces is equal to the change in kinetic energy of the system.

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[example] As the figure shows, an object A resting initially moves down and another object B ascends. Calculate the speed of A and B when the distance that A moves is S .(neglecting the friction and mass of the pulley) Solution: Choose A,B and the rope as a system we study. Forces on the system is shown in the figure. Work done by gravity GA of A is positive, work done by gravity GB of B is negative, and the normal force N doesn’t do work, the total work done by external forces is given by

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Net work done by internal forces TA and TB is zero, that is, According to the theorem of kinetic energy,we have Solving for v, we obtain The initial kinetic energy of the system is zero and the final is given by

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13.3.3 principle of work and energy From the theorem of kinetic energy for system Work done by external forces and internal non-conservative forces on the system is equal to the change in mechanical energy.

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Solution: (1) The coordinate that we set up is shown in the figure. Suppose that at some instant length of the droopy part of the chain is x .

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(2)apply the theorem of kinetic energy to solve for speed v

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(2)apply the principle of work and energy to solve for speed v Choose a system composed of the chain, the table and the earth. Let the surface of the table to be the ground point. Therefore no external force acts on the system. initial state: final state: The work done by the friction as initial non-conservative force is given by

Slide26: 

From the theorem of work and energy We have Solving for v, we obtain

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13.3.4 conservation of mechanical energy according to the theorem of kinetic energy for a particle, we have Apply it to every particle in a system, we can get By the definition of potential energy, we have The work done by internal forces consists of two parts. One is done by conservative internal forces, and the other is done by non-conservative internal forces. That is,

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Then equation (2) becomes By adjustment, we have From the equation above, we can see that if then ----the law of conservation of mechanical energy

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[example] A steel ball of mass 1kg is attached to one end of a rope of length l . The other end of the rope is fixed. The rope is pulled aside to the horizontal and released. At the lowest point of the swing elastic collision occurs between the ball and a block of mass 5kg resting on the rough surface. After the collision the ball can reach a vertical height of h=0.35m, and the block slides horizontally and finally stops somewhere. Find (1)the length of the rope;（2）the work done against the friction.（ g = 10m/s2 ） solution：

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13.4 potential energy and stability For simplicity, we assume that an object moves only in the x direction, and let x=0 be the point where there is no force. In addition, we suppose that the force acting on the object is proportional to how far the object is from the point x=0: Where b is some constant. The total work we do moving the object from 0 to x is the integral Where s is a dummy variable. Since F(s)=bs, this integral is equal to

Slide31: 

The object’s potential energy changes by the same amount , so we have Let U(0)=0, we get the potential energy with respect to the level of reference x=0 The graph of U(x) is a parabola which opens downward if b is positive and upward if b is negative.

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The slope of a tangent to a curve is equal to the derivative at that point. The derivatives of the curves are Two significant features can be seen from our analysis Maximum or minimum A restoring force

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The maximum or minimum of the potential energy occurs at just the point where dU/dx=0, and therefore at just the point where the object has no force on it. We say that the object is in equilibrium state. The place where such equilibrium occurs, where dU/dx=0, is called the equilibrium position. There are three types of equilibrium: (1)stable equilibrium Potential energy is a minimum . (2)unstable equilibrium Potential energy is a maximum . (3)neutral equilibrium Potential energy is constant .

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We know that a minimum or maximum of potential energy will occur at values of r that satisfy Taking the derivative, we see that this condition implies

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Solving for r, we find that r=r0 is the equilibrium separation distance. The value of the potential energy at this point is U(r0)=-U0 . assignments P252: 6, 7; P263: 20; P283: 16, 17, 19; P375: 14, 17,