Classical Mechanics Virtual Work & d'Alembert's Principle Dipan Kumar Ghosh UM-DAE Centre for Excellence in Basic Sciences Kalina, Mumbai 400098 August 15, 2016 1 Constraints Motion of a system of particles is often constrained, either geometrically or kinematically. Constraints reduce the number of degrees of freedom of a given body. Consider the motion of a single particle in space. For free, unconstrained motion, it has three degrees of freedom which are usually expressed by three coordinates such as x; y; z or r; θ; ' etc. If, however, the particle is constrained to move on the surface of a sphere, we must have (taking Cartesian coordinates), x2 + y2 + z2 = R2 which reduces the number of degrees of freedom by one. Consider two masses connected by a rigid rod, like a dumbbell. Two particles have 6 degrees of freedom, Since the distance between the two bodies remains constant, we have the constraint 2 2 2 2 (x1 − x2) + (y1 − y2) + (z1 − z2) = d which reduces the degree of freedom to 5. These are examples of geometric or holonomic constraints whoch are expressible as algebraic equations involving the coordinates. There are other constraints which restrict the motion of bodies . Some of these are expressible as differential equations which constrain the coordinates and components of velocities. These are called kinematic constraints. Non-integrable kinematic constraints which cannot be reduced to holonomic constraints are called non-holonomic constraints. Thus, if m is the dimension of the configuration space (i.e., the number of generalised coordinates), holonomic constraints are expressible as equations of the form i f (t; q1; q2; : : : ; qm) = 0; 1 ≤ i ≤ k 1 c D. K. Ghosh, IIT Bombay 2 where k is the number of constraints. Holonomic constraints are called scleronomic if they do not explicitly depend on time. Time dependent constraints are called rheonomic. Kinematic constraints are expressed as equations in the phase space i f (t; q1; q2; : : : ; qm;_q1; q_2;:::; q_m) = 0; 1 ≤ i ≤ k Both the constraints are classified as rheonomic if they explicitly depend on time. Sometimes a constraint may appear to be kinematic but may be in reality holonomic. For instance, a constraint of the type Ax_ + B = 0 may actually be holonomic if there exists a function f such that A = @f=@x and B = @f=@t. We then have, @f df @f df = + @x dt @t which gives a holonomic constraint f = constant. Example 1: Consider two masses connected by pulleys, as shown. In general two particles have 6 degrees of freedom. However, m1 can only move along the x direction and m2 along the z direction. Thus y1 = z1 = 0 and x2 = y2 = 0. We are left with two degrees of freedom. However, if m1 moves along the x direction by a distance d, m2 would have to move along z direction by 2d, i.e., we get another holonomic constraint, z2 − 2x1 = 0 which reduces the degree of freedom further by one. The problem is essentially a one dimensional problem. m 1 P 1 z x P 2 m 2 Non-holonomic constraint could be in the form of differential equations or algebraic inequalities. For instance, a particle constrained to move inside a sphere of radius R satisfies x2 + y2 + z2 < R2. Consider a disk rolling on a horizontal surface, on x-y plane. c D. K. Ghosh, IIT Bombay 3 z y P φ θ x If the disk is rolling without slipping, keeping its plane vertical, we need four coordi- nates to describe the position of the disk. These are the x and y coordinates of the centre of the disk, the angle ' by which a fixed point on the rim of the disk has rotated about the axis of rotation and the angle θ that the axis of the disk makes with the x axis. If R is the radius of the disk, the velocity of the disk is given by v = R! = R'_ (1) Since the disk remains vertical, the components of the velocity of the centre of the disk are given by dx = v sin θ (2) dt dy = −v cos θ (3) dt Using eqs. (1) to (3) we get dx d' = R sin θ dt dt dy d' = −R cos θ dt dt the minus sign is due to the sense of rotation being opposite to the positive angle. Com- bining these two we get the following pair of differential equations: dx = R sin θd' dy = −R cos θd' These equations cannot be further reduced and we cannot connect x; y; θ and ' by an algebraic equality, showing that the constraint is non-holonomic. c D. K. Ghosh, IIT Bombay 4 Tackling problems with non-holonomic constraints is more difficult and no general pre- scription can be provided for their solution. Constraints introduce two new elements into problem solving. Since the generalised coordinates are no longer independent, the equations of motion are not independent either. Constraints arise from forces between elements whose nature is unknown. These forces are known only by the effect they have on motion of the system. 2 Virtual Displacement A real displacement of particles constituting a system happens over a finite time. Such a displacement of, say, the i-th particle is, in general, function of all the generalised coordinates as well as of time. If the position of the i-th particle is written as ~ri = ~ri(q1; q2; : : : ; qs) The total differential of the position vector is then written as s X @~ri @~ri d~r = dq + dt (4) i @q j @t j=1 j A virtual or an imagined displacement is instantaneous and is consistent with the con- straints on the system. The displacements, whether real or virtual, result in admissible geometrical configuration of the system. In case of a virtual displacement, we have, s X @~ri d~r = δq (5) i @q j j=1 j where we have use δq to indicate a virtual displacement while reserving dq for real dis- placement. For a real displacement, the forces of constraints may change. The velocity is given by s d~ri X @~ri @~ri ~r_ = = q_ + (6) i dt @q j @t j=1 j so that we have, ~ s @r_i X @~ri @~ri = δ = (7) @q_ @q jk @q k j=1 j k (Note the structure of the above equation - as if the dots cancel!) 2.1 Principle of Virtual Work Consider a system of N particles under time dependent holonomic constraints. If q1; : : : ; qs be a set of generalized coordinates, the virtual displacement of the i-th particle is given c D. K. Ghosh, IIT Bombay 5 by eqn. (5). Suppose the system, under the action of applied forces as well as those of constraints, is in equilibrium. The total force acting on each particle is then zero, ~ Fi = 0; (i = 1;:::;N) ~ We then have, for the virtual work done by Fi in the displacement δ~ri is ~ Fi · δ~ri = 0 so that the total work done is X ~ δW = Fi · δ~ri = 0 i The total force acting on any particle can be split into two: an applied part and a part due to the constraints, ~ ~ a ~ c Fi = Fi + Fi then we have X ~ a X ~ c δW = Fi · δ~ri + Fi · δ~ri = 0 (8) i i The forces of constraints (e.g. normal reaction, tension, rigid body constraints etc.) do not do any work. This is general true of scleronomic holonomic constraints and this statement is central hypothesis in the principle of virtual work. Two examples illustrate the hypothesis. Consider two types of displacements consistent with the constraints on a rigid body. A displacement along the line joining two particles does not do any net work because the reactions are equal and opposite. In order to be consistent with rigid body constraints, for a pair of particles j and k, we must have δrk = δrj. The work done is fkj · δrj + fjk · δrk = (fkj + fjk) · δrj = 0 where we have used δrj = δrk. Thus there is no work done. The second type of displacement is along the arc of a circle normal to the line joining the particles.As the forces of constraints are normal to the direction of displacement, the work done is once again zero. Consider the pulley arrangement in Example 1. When m1 moves by an amount δx to the right, m2 moves by 2δx downwards in order to keep the sum of the lengths of the two ropes constant. The only applied forces are the gravity and friction. Thus by the principle of virtual work, we have, −µmgδx + mg2δx = 0 which shows that for static equilibrium m2 = µm1=2. Example 2: Two frictionless blocks of mass m each are connected by a massless rigid rod. The system is constrained to move in the vertical plane. c D. K. Ghosh, IIT Bombay 6 mg R δx 1 mg x θ δ 2 F N If the block on the vertical track undergoes a virtual displacement δx1 and that on the horizontal plane has a virtual displacement δx2, we have δx1 sin θ = δx2 cos θ which is the constraint which keeps the rod length constant. The gravity does work on the mass on the vertical track while the applied force F2 is responsible for work on the horizontally moving block, mgδx1 + F2δx2 = 0. Thus we have δx1 F2 = −mg = −mg cot θ δx2 Example 3: Consider an Atwood's machine in equilibrium. In this case we have the constraint y1 + y2 = l = constant. m m 1 2 Thus we have δy1 = −δy2.