Question

Figure 4-28 shows four tracks (either half- or quarter-circles) that can be taken by a train, which moves at a constant speed. Rank the tracks according to the magnitude of a train’s acceleration on the curved portion, greatest first.

Short answer

$\mathrm{Rank}:2>1=4>3$

Step-by-step solution

Given information

Rank according to radius: 3 > 4 = 1 > 2

To understand the concept

This problem is based on concept of centripetal acceleration due to motion on curved path.

Formulae:

Centripetal acceleration is given by,

$\left(a_c\right)=\frac{v^2}{r}$

The rank of the tracks according to the magnitude of train’s acceleration on the curved portion.

Centripetal acceleration can be written as,

$\mathrm{a}=\frac{{\mathrm{v}}^2}{\mathrm{r}}$

Distance between the right side and the dotted lines represents the radius of the curved portion of the track. Radius of the curved portion of the track is smallest for track 2. For track 1 and 4, radius is same, but it is greater than track 2 and smaller than track 3. Radius is greatest for track 3.

As speed is same for all tracks, centripetal acceleration is inversely proportional to radius of curve.

Hence, ranking according to acceleration is:

$\mathrm{Rank}\mathit{:}\mathit{2}\mathit{>}\mathit{1}\mathit{=}\mathit{4}\mathit{>}\mathit{3}$