NCERT Solutions

Point Object Approximation in Motion Examples

Concept: Point Object Approximation

The point object approximation is used in kinematics when the linear dimensions of a body are much smaller than the distance it covers during its motion. Mathematically, if $L$ is the characteristic length of the object and $s$ is the distance traveled:

$$s \gg L$$

Given:

• (a) Railway carriage between stations.
• (b) Monkey on a cyclist on a circular track.
• (c) Spinning cricket ball turning on the ground.
• (d) Tumbling beaker falling off a table.

Solving:

1. Railway Carriage:

   The length of a carriage is typically $20\,\text{m}$ to $30\,\text{m}$, while the distance between two stations is usually several kilometers ($10^{3}\,\text{m}$ to $10^{4}\,\text{m}$). Since the distance $s$ is much greater than the length $L$, the carriage can be considered a point object.

2. Monkey on a Cyclist:

   The circumference of a standard circular track is much larger than the physical size of a monkey or a human. Because the distance covered $s = 2\pi R$ is significantly greater than the size of the monkey, it can be considered a point object.

3. Spinning Cricket Ball:

   When a ball spins and turns "sharply," the rotational motion and the contact mechanics with the ground are critical. The distance over which the turn occurs is comparable to the ball's diameter. Therefore, its size cannot be ignored, and it is not a point object.

4. Tumbling Beaker:

   A table height is approximately $1\,\text{m}$, and a beaker is roughly $0.1\,\text{m}$ tall. The distance of the fall is not significantly larger than the object's dimensions. Furthermore, "tumbling" implies rotational motion where different parts of the beaker move differently. Thus, it is not a point object.

Answer

$$\text{(a) and (b)}$$