Section 14–2 Constrained motion

(Frictionless constraints / Constraint forces / Analysis of constraint forces)

In this section, Feynman discusses frictionless constraints, constraint forces, and analysis of constraint forces.

1. Frictionless constraints:

“…suppose that we have a sloping or a curved track, and a particle that must move along the track, but without friction... These are examples of what we call fixed, frictionless constraints (Feynman et al., 1963, section 14–2 Constrained motion).”

Feynman gives two examples of fixed and frictionless constraints: (1) a particle moves along a sloping or a curved track that is frictionless; (2) a string constrains a weight to move circularly about a pivot point. He adds that the pivot point can be changed by having the string hit a peg such that the path of the weight is along two circles of different radii. Alternatively, Landau and Lifshitz (1976) explain that “[i]t is often necessary to deal with mechanical systems in which the interaction between different bodies (or particles) takes the form of constraints, i.e., restrictions on their relative position. In practice, constraints are effected by means of rods, strings, hinges and so on (p. 10).” Generally speaking, the word constraints can be classified as two categories: (i) constraints due to forces, and (ii) constraints due to equations (Waghmare, 1990). Physicists do not always conceptualize constraints as physical objects.

Some textbook authors describe at least four kinds of constraints: (1) Holonomic: the constraint on the positions of a system of particles, e.g., a particle is constrained to move on a plane: Ax + By + Cz = D. (2) Nonholonomic: constraints that restrict the velocities of particles, e.g., av₁ + bv₂ + cv₃ + … = N. (3) Scleronomous: constraints that do not depend on time explicitly, e.g., a pendulum of inextensible string can be described by the equation, x² + y² = l². (4) Rheonomous: constraints that specify time explicitly, e.g., a pendulum can be described by the equation, x² + y² = l²(t) where l(t) is the length of a string at time (t). Feynman would complain about these terms and possibly say velocity-independent (holonomic), velocity-dependent (nonholonomic), time-dependent (rheonomous), and time-independent (scleronomous).

2. Constraint forces:

“In motion with a fixed frictionless constraint, no work is done by the constraint because the forces of constraint are always at right angles to the motion (Feynman et al., 1963, section 14–2 Constrained motion).”

According to Feynman, “forces of constraint” mean forces which are applied to an object directly by a constraint itself. It may be a contact force with a track or a tension in a string. During an object’s motion with a fixed and frictionless constraint, there is no work done because the forces of constraint are always at right angles to the motion. However, some physicists may criticize Feynman’s definition of constraint forces. For example, we should not assume that frictional forces are definitely not forces of constraint. Note that static frictional forces can constrain an object to move circularly and the work done by static frictional forces is also zero.

We may find some of the following features in a more comprehensive definition of constraint forces: (1) It may be a contact force, tension, or static frictional force (non-sliding). (2) The constraint force on a particle could be exerted by a rod, string, or surface of a track. (3) It restricts the path of a particle or the motion of a system of particles. (4) It limits a particle’s freedom of motion under certain geometric or kinematic conditions. (5) The work done by a constraint force is zero because the force is perpendicular to the motion of the particle (W = 0). On the other hand, one may relate constraint forces to D’Alembert’s principle of virtual work.

3. Analysis of constraint forces:

“…the work done by the net force is equal to the sum of the works done by all the parts into which we have divided the force in making the analysis (Feynman et al., 1963, section 14–2 Constrained motion).”

Feynman explains that the work done by the resultant force can be analyzed as the sum of the works done by various “components” of forces. For instance, we can analyze the forces in terms of the x-component of all forces and the y-component of all forces, or using other co-ordinate systems. The analysis of constraint forces in this section appears simple because the constraints are idealized as fixed and frictionless. Physics teachers should add that the analysis involves mathematical skills that can be improved by practice. In some mechanical problems, it is either difficult or even impossible to determine the constraint forces on the objects.

There are at least two types of difficulties in solving mechanical problems that are related to constraints: “First, the co-ordinates r_i are no longer all independent, since they are connected by the equations of constraint: hence the equations of motion are not all independent. Second, the forces of constraint, e.g., the force that the wire exerts on the bead (or the wall on the gas particle), is not furnished a priori (Goldstein, Poole, and Safko, 2001, p. 13).” Simply put, some mechanical problems have many constraint forces and require as many independent equations of constraint that help to solve the equations of motion, provided it is solvable. In essence, the constraint forces are related to the motion of the particles in the system, and they can be calculated after the motion has been determined (or use F = -dV/dr for spherical co-ordinates).

Questions for discussion:

1. How would you classify the different types of constraints?

2. How would you criticize Feynman’s definition of constraint forces?

3. How would you determine the constraint motion or constraint forces?

The moral of the lesson: during the motion of a particle with a fixed and frictionless constraint, there is no work done by the constraint because the constraint forces are perpendicular to the motion.

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

2. Goldstein, H., Poole, C. P., & Safko, J. L. (2001). Classical Mechanics (3rd ed.). Massachusetts: Addison-Wesley.

3. Landau, L. D., & Lifshitz, E. M. (1976). Mechanics (3rd ed.). Oxford: Pergamon Press.

4. Waghmare, Y. R. (1990). Classical Mechanics. New Delhi: Prentice Hall.