Chapter 8

Constrained Motion

In many cases, the object we are studying is not free to move in every direction. For example, a bead on a wire can only move along the wire. The shape and position of the wire determine the path of the bead, but the bead is still free to move in many different ways along that wire. In this case we call the motion “constrained”. The constraints can be strong, by restricting the motion to be along a given path, or the constraints can be weak, such as for a bead caught between two parallel glass plates, or for a car driving on the terrain. Constraints may also arise because objects are connected to each other: The motion of an individual atom in a large rigid molecule is constrained by the motion of the whole molecule, and a small part of a spinning wheel is constrained to follow the motion of the wheel.

We have learned that the motion of an object can be determined from the forces acting on the object. This is, of course, also the case for constrained motion. But in many cases it is not practical to include the forces that restrict the motion, either because we do not have good models for them, or because we would rather like to determine these constraining forces from what we know about the motion—from the fact that the object follows a particular path. For example, a bead moving along a wire will be affected by normal forces from the wire on the bead, and it is these forces that cause the bead to follow the path given by the wire. However, we may not want to model these forces in detail, instead we are interested in the consequences of constraints—we want to find the normal force when we know that the bead follows the wire.

In this chapter, we will discuss constrained motion. Starting from the simplest case of motion constrained to a line, through circular motion, to motion constrained to be along a general curve.

8.1 Linear Motion

The simplest case of a constrained motion is that of a bead moving along a straight, rigid wire (see Fig. 8.1) or a car moving along a straight road. We call such a motion linearly constrained—because the motion is constrained to follow a given line.

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215

A. Malthe-Sørenssen, Elementary Mechanics Using Python,

Undergraduate Lecture Notes in Physics, DOI 10.1007/978-3-319-19596-4_8

216	8 Constrained Motion

(A)	(b)
	Y
z	x
	’
	Y	’
		x
	S’

Y R

S x

x’

θ r

Fig. 8.1 a The motion of a bead moving along a straight wire is constrained to be along the line described by the wire. b The motion of a small train moving up along an inclined track is also constrained by the track. The motion can be described both in a coordinate system with axes along the track, and in a coordinate system with axis in the horizontal and vertical direction

How can we formulate a mathematical constraint corresponding to the motion of a bead on a wire as in Fig. 8.1? The simplest way is to choose the x -axis to be along the wire. Then the bead can only move along the x -axis, and the y-, and z-components remain zero throughout the motion:

r(t ) = x (t ) i + 0 j + 0 k = x (t ) i .

(8.1)

Because we are free to choose the coordinate system, we can always make such a choice for linear motion.

We can also formulate the constraint without fixing the x -axis to be along the direction of motion, such as for the train in Fig. 8.1. The train moves along a slope that forms the angle θ with the horizon. We can describe the slope by the point R and the unit vector uˆ = cos θ i + sin θ j which points along the slope. The position of the train is then given by how far along the slope it has moved, s(t ):

r(t ) = R + uˆ s(t ) .

(8.2)

We call s(t ) the distance, and notice that it would correspond to the coordinate x ′ of a coordinate system oriented along the slope. Since R is a constant, the velocity of the train is

d r	=	d R	+uˆ	d s	= uˆ	d s	,	(8.3)
d t		d t		d t		d t

=0

where d s/d t is the velocity measured along the track, which is what you would measure from a speedometer. The distance s(t ) traveled reflects the motion of the bead. It does not have to be only increasing, but may reflect a complicated motion along the wire.