How to determine whether an object is a point object?

Question

I know that we can consider an object as point object, if its size is negligible as compared to distance traveled by it in reasonable amount of time. But in my book Ncert there is questions which asks to determine which of the following are point objects:
(a) a railway carriage moving without jerks between two stations.
(b) a monkey sitting on top of a man cycling smoothly on a circular track.
(c) a spinning cricket ball that turns sharply on hitting the ground.
(d) a tumbling beaker that has slipped off the edge of a table

It states that (a) and (b) can be considered as point objects but (c) and (d) cannot. Why cannot we consider them as point object if we do not the distance they have travelled and their size?

Answer 1

I know that we can consider an object as point object, if its size is negligible as compared to distance traveled by it in reasonable amount of time.

This is not always correct. What is really meant by a point-like object (i.e., and object to which we can apply the Newton's laws) is an object whose internal motion can be ignored for the practical purposes (i.e., in a particular problem.)

When we interested in the behavior caused by the object spinning or oscillating, i.e., by the relative motion of the points within the object itself, we cannot treat it as a point-like - it is just a logical contradiction.

That the size is negligible in comparison to the distance traveled is meaningful in the context of extended objects, as a way of avoiding ambiguity about the object position: e.g., a hundred meters long train moving a distance of one millimeter is a poorly defined problem. In some situations we could still associate the position of the object with a particular point in it or its center-of-mass, but if the relative displacement of the points within the object (e.g., the train carriages) is bigger than its displacement as a whole this is still problematic.

In practice the first step is often to consider the motion of the object as a whole as the motion of its center-of-mass, with the other motions superimposed on it - the rotation of the rigid body, vibrations of a molecule, statistical physics description of an ideal gas.

Answer 2

In my opinion the question from the book is a very poor question. Whether a real world object can be treated as a point object or not depends on which aspects of the object and its motion we are interested in. In (c) and (d), since attention is drawn to the rotation of the object ("spinning" in one case and "tumbling" in the other) then we are presumably not meant to treat them as point objects. However, (a) and (b) are ambiguous.

Taking (a) as an example, if we are only interested in the speed of the railway carriage then we can treat it as a point object since we can assume all of its parts will be travelling at the same speed. However, if we are interested in the minimum separation between parallel railway lines so that two trains can safely pass each other on a curve with given radius, or the best design of points to allow a carriage to pass across them without discomfort to passengers then we cannot treat the carriage as a point object - we need to take into account its length and the position of its wheels.

Answer 3

If for the purposes of a given problem the only relevant energy associated with a body is the kinetic energy of its center of mass, then you can think of it as a "point object" (the "point" having a mass equal to the total mass of the original object, and being located at the position of the center of mass of it).

Answer 4

The question cannot be answered without some more context. For example you could assume that a railway carriages is a point object at its centre of mass perhaps in determining a displacement, velocity or acceleration of the centre of mass but if you needed to know whether or not a carriage would topple if went round a corner at a certain speed then that could not be done by just considering the carriage as a point object.

Now in the question there is some context for some of the responses and in particular for c and d there is mention of rotation of an object which is problematic for a point object.

Suggestion b is problematic which most have experienced. Does one not try and lean into a corner with forces not all acting at the centre of mass.

It is a pity that the OP has been subjected to such an ill-considered question.

Answer 5

A geometric point doesn’t spin or tumble. Only extended objects do. Newton’s second law $F=ma$ applies to a point object or to the center of mass of an extended object where all the mass can be considered located at a point. For rotational motion, which only applies to an extended object, the version of Newton’s second law that applies is $\tau =I\alpha$

Hope this helps

Answer 6

However, the question supplies limited information, motion is a relative concept and such physical problems can be idealized depending on the objective.

Generally, point objects have dimensions that are too small to interfere with the effect of forces applied. For example: The motion of an object due to action of a force in a way that it does not contribute to rotation (i.e. contributing only to translatory or straight-line motion) can be "safely" considered as motion of a point object. In such a case the dimension of the object from its centre of mass is very small or the force itself acts on the centre of mass of the object.

Let us look at the mentioned problems one by one.

a). It is a simple case of translatory motion, the coaches of the train will not turn over about its centre of mass (that it would have been due to jerks). The motion of the train can be easily determined by motion of a point attached anywhere on the train.

b). Case of circular motion abount a centre. The motion of both the monkey and the cyclist about the centre can be considered as motion of a point attached anywhere on them (the only difference that may arise in the results will be due to thickness of them and that can again be neglected for larger radius of motion).

c). Here, the spinning (turning) of the ball is considered along with the motion of its centre of mass. The spinning of the ball will depend on the interaction of its surface and the ground (due to friction). The forces that are not acting at the centre of mass of the ball will result in turning (the forces are acting at many other points inside the ball that are moving at different velocities). These "many points" are point objects forming the ball.

d). Here, the rotation can be justified by considering motion of "many points" while those interested only in the downfall can consider only the motion of centre of mass (a point object falling under the influence of gravity; neglecting drag; considering the downfall as motion of a pint object makes more sense but I think that the book wants you to focus on the fact that rotation is a property that is specific to a many point systems, each point having different velocity).

Don't overthink the problem. Obviously, trains are not point objects, neither are men nor mankeys but formulation of a physical problem might let us think so.