1. MA4248 Weeks 6-7. Topics Work and Energy, Single Particle Constraints, Multiple Particle Constraints, The Principle of Virtual Work and d’Alembert’s Principle 1 Work and Energy: the English words originated, via Germanic and Greek branches, respectively, from the Proto Indo-European word Werg about 7 k years ago my homepage under courses/Ussc2001/Energy1.pdf The work done on a particle that is displaced by in a constant force fieldequals This work has units of energy.

2. CONSTANT FORCE FIELDS Let us consider this situation in detail. Let 2 Thenhence and if particle whose trajectory is then construct the function denote affine space and choose a point by is constantthen since Newton’s second law implies that is the net force on a

3. CONSTANT FORCE FIELDS and 3

4. CONSERVATIVE FORCE FIELDS 4 The argument on page 20 in Week 1-3 Vufoils show that this total energy is constant for any conservative force field. Let us consider the following converse There exists a force field F and a function U that are functions on A (time independent) such that is constant for every particle of mass m that moves with net force F. Then henceis conservative.

5. WORK OVER A SMOOTH CURVE 5 Thomas, p. 1062. The work done by a force (field) over a smooth curve parameterized by a smooth vector valued functionon the interval [a,b] is If is conservative then

6. PATH INDEPENDENCE AND COMPONENT TEST 6 Thomas, p. 1072 A vector fieldis conservative depends only on the endpointsif and only if andof the curve. Thomas, p. 1074 is conservative if and only if

7. SINGLE PARTICLE CONSTRAINT 7 In Tutorial 3, Prob. 5 you computed the trajectory of a ring sliding down a straight rod by assuming either 1.that the total energy is conserved, or 2.that the force of constraint is orthogonal to the rod Why do these assumptions yield the same trajectory? Consider a particle having mass m that is constrained to move along a curve parameterized by a function of a variable s, called a generalized coordinate. This may be the case if the particle consists of a ring that slides along a rigid wire. We will first assume that the curve does not move so that it is independent of time

8. SINGLE PARTICLE CONSTRAINT 8 Therefore, the trajectory of the particle must equal where t denotes time and s(t) is a function of time Newton’s second law implies that whereis the applied force that would be there if the physical constraint (wire) was removed, and is the force of constraint defined by this equation

9. SINGLE PARTICLE CONSTRAINT 9 Define over the curveThe work performed by is

10. SINGLE PARTICLE CONSTRAINT 10 Then is the change of kinetic energy over the time interval

11. SINGLE PARTICLE CONSTRAINT 11 If the applied force is conservative and hence is the change in total energy It equals zero for all time intervals iff

12. SINGLE PARTICLE CONSTRAINT 12 In this case, it is very convenient to use arc length parameterization of the curve, then and energy conservation implies that the trajectory is determined, up to the initial position, by the first order differential equation

13. SINGLE PARTICLE CONSTRAINT 13 We now consider the case where the particle is constrained to move along a curve that is moving with time as in Tutorial 5, Problems 4 and 5. In this case the force of constraint may perform work on the particle, yet it is reasonable to assume that at each value of time the force of constraint is orthogonal to the curve described at that time. Note: the curve at a specific time does not describe the actual trajectory of the particle, this very important fact is illustrated in Fig. 2.04 on page 31 of the textbook.

14. SINGLE PARTICLE CONSTRAINT 14 Therefore, the trajectory of the particle must equal since we now have a time varying family of curves. Newton’s second law implies that and the orthogonality condition implies that There are 4 unknowns and 4 equations

15. SINGLE PARTICLE CONSTRAINT 15 We now consider a particle that is constrained to move along a (possibly moving) surface. Then the trajectory is determined by the principle that constraint force at time t is orthogonal to the constraint surface at time t Parameterize the surface at time t is by a (possibly time varying) function of generalized coordinates so that at time t Newton implies

16. SINGLE PARTICLE CONSTRAINT 16 Note that we now have 5 unknowns, the 2 generalized coordinates and 3 components of the constraint force. Newton gives us three equations. We need 2 more. They are provided by the orthogonality principle:

17. SINGLE PARTICLE CONSTRAINT 17 If the applied force is conservative and if the surface is independent of time then energy is conserved. Energy conservation is not sufficient to determine the motion since it provides only 1 additional equation. If the surface is moving then the forces of constraint may (and usually do) perform work since since

18. SINGLE PARTICLE CONSTRAINT 18 In the previous discussion of a single particle, we used generalized coordinates. However, we could have used rectangular coordinates. If we did then we would have 6 unknown variables – 3 coordinates x, y, z for the position of the particle and 3 coordinates of the force of constraint. These 6 variables are determined (by the solution of differential equations) by the constraint equations (1 for a surface and 2 for a curve), Newton’s second law (3 equations), and the principle that the force of constraint is orthogonal to the constraint set (2 for a surface, 1 for a curve)

19. MULTIPLE PARTICLE CONSTRAINTS 19 For a system with N particles, Newton’s 2 nd law gives 3N equations in 6N variables. We need 3N more! M=3N-f holonomic constraints, given by equations give a total of 6N-f equations, we need f more! These f equations will be provided by the Principle Of Virtual Work. For 1 particle, this principle says that the force of constraint is orthogonal to the surface (M=1) or curve (M=2) that the particle moves on.

20. MULTIPLE PARTICLE CONSTRAINTS 20 For multiple particle constraints virtual displacements are any displacements that satisfy The principle of virtual work says that the total work done by the forces of constraint over these displacements equals zero

21. GENERALIZED COORDINATES 21 The set of virtual displacements form a vector space having dimension f = 3N-M, these are the number of degrees of freedom of the system If we introduce generalized coordinates then and we can find a basis for this vector space

22. GENERALIZED COORDINATES 22 Then the principle of work can be expressed by This holds for all choices of iff These are the f additional equations that we require.

23. D’ALEMBERT’S PRINCIPLE 23 Then the principle of work can be expressed by The work done by the applied forces, plus the work done by the inertial forces, in a virtual displacement is zero

24. EXAMPLES 24 Two particles connected by a light rigid rod (pp.29-30) The forces of constraint are proportional to The constraintimplies that and Newton’s second law implies thathence

25. EXAMPLES 25 Inclined plane (pp.34) If the block undergoes a virtual displacement Inclined plane p. 34 down the plane then the applied force, gravity, does work generalized coordinate S is the distance down the inclined plane and the inertial force (oriented up the plane) which yields the well-known resultdoes work