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Chapter 3 Work and Energy. §3-1 Work. §3-2 Kinetic Energy and the Law of Kinetic Energy. §3-3 Conservative Force, Potential Energy. §3-4 The Work-Energy theorem Conservation of Mechanical Energy. §3-5 The Conservation of Energy. §3-1 Work.
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§3-1 Work §3-2 Kinetic Energy and the Law of Kinetic Energy §3-3 Conservative Force, Potential Energy §3-4 The Work-Energy theorem Conservation of Mechanical Energy §3-5 The Conservation of Energy
§3-1 Work --variable force 1.Work Equal to the displacement times the component of force along the displacement. -- element work or ab：
In Cartesian coordinate system 2.Work done by resultant force If Then The work done by the resultant force = the algebraic sum of the works done by every force.
3. Power The work done per unit time 4.Work done by action-reaction pair of forces
relative displacement 与参考点的选择无关
§3-2 Work-kinetic energy theorem 1. WKE Theo. of a particle
Definition -- Kinetic energy The total work done on a particle = the increment of its kinetic energy --Work-kinetic energy theorem ( the Law of kinetic energy)
2. WKE Theo. of particle system According to above For m1 … … For m2 +
-- Work done by internal force -- Work done by external force Final KE Initial KE
Extend this conclusion to the system including n particles The sum of the works done by all external forces and internal forces = the increment of the system’s KE. -- System’s work-kinetic energy theorem
v · · R [Example] A particle with mass of m is fixed on the end of a cord and moves around a circle in horizontal coarse plane. Suppose the radius of the circle is R. And vo vo/2 when the particle moves one revolution. Calculate The work done by friction force. frictional coefficient. How many revolutions does the particle move before it rests?
Solution According to WKE theo., Opposite to the moving direction We get
(rev) Suppose the P moves n rev. before it rests. According to work-kinetic energy theorem, We have
1. Conservative force The work done by Cons. force depend only on the initial and final positions and not on the path. §3-3 Conservative force Potential energy The integration of Cons. force along a close path l is equal to zero. Otherwise, non-conservative force The potential energy can be introduced when the work is done by the Cons. Force.
2. Potential energy (1) PE of weight Gravitational force or
--PE of weight Definition then the work done by GF = the reduction of PE of weight
If then PE of weight at point a=the work done by GF moving m from a to zero PE point. The point of zero PE of weight is arbitrary
Elastic force (2) Elastic PE Definition --Elastic PE
then the work done by EF = the reduction of elastic PE The point of zero elastic PE： relaxed position of spring (x=0)
(3) Universal gravitational PE Universal gravitational force
when Definition ----UGPE then The point of zero UGPE： the distance of both particles is infinity( r
Remarks The PE of a particle at a point is relative and the change of a particle from one point to another point is absolute. Only conservative force can we introduce potential energy. The done by conservative force = the reduction of PE
Gravitational force Elastic force Universal gravitational force PE belongs to the system. Conservative internal force The frictional force between bodies is non-conservative internal force
§3-4 The work-energy theorem Conservation of Mechanical Energy System’s work-kinetic energy theorem Internal force =Conservative IF+non-Cons.IF
Let -- mechanical energy of the system The sum of the work done by the external forces and non-conservative forces equals to the increment of the mechanical energy of the system from initial state to final state. -- the work-energy theorem of a system
when We have --Conservation of mechanical energy
F [Example] Two boards with mass of m1，m2 (m2>m1) connect with a weightless spring. If the spring can pull m2 out of the ground after the F is removed, How much the F must be exerted on m1 at lest? How is about the result if m1，m2 change their position?
Suppose the length of the spring is compressed as the F is exerted. And m2 is pulled out of the ground as the length is just stretched after the F is removed Solution then
Two boards+spring+earth = system Its mechanical energy is conservation Chose the point of zero PE : The spring is free length ( no information) We can get The result do not change if m1，m2 change their position.
§3-5 The Conservation of Energy Friction exists everywhere
The frictional force is called as a non-conservative force or a dissipative force which exists everywhere. Its work depends on the path and it is always negative. So if the dissipative forces exist such as the internally frictional force, it is sure that the mechanical energy of the system decreases. According to the work-energy theorem
The decrease of mechanical energy is transformed into other kinds of energy such as heat energy because of friction. Which leads to the increase of temperature of system so that the internal energy of the system has an increment. In order to simplify this problem, if we suppose Wex=0 We have
Re-write above formula The change of internal energy + the change of mechanical energy = conservation So we can get the generalized conservation law of energy as follow Energy may be transformed from one kind to another in an isolated system. But it cannot be created or destroyed. The total energy of the system always remains constant.
