## Statement

A toy train nicknamed "torpedo" moves in a circular trajectory of 2 m radius without possibility of changing its linear velocity. Knowing it takes 10 seconds for a full rotation, calculate:

a) Its angular and linear velocities.

b) The angle and the distance traveled in 2 minutes.

c) Its acceleration.

## Solution

We have a uniform circular motion because the trajectory is a circumference and the speed does not change throughout the motion.

**Question a)**

**Data**

R = 4 m

T = 10 s

**Resolution**

To calculate the angular velocity, we will use the following expression:

$$\omega =\frac{2\xb7\pi}{T}\Rightarrow \phantom{\rule{0ex}{0ex}}\omega =\frac{6.28rad}{10s}\Rightarrow \phantom{\rule{0ex}{0ex}}\overline{)\omega =0.628\raisebox{1ex}{$rad$}\!\left/ \!\raisebox{-1ex}{$s$}\right.}$$

And the linear velocity is:

$$v=\omega \xb7R\Rightarrow \phantom{\rule{0ex}{0ex}}v=0.628\raisebox{1ex}{$rad$}\!\left/ \!\raisebox{-1ex}{$s$}\right.\xb72m\Rightarrow \phantom{\rule{0ex}{0ex}}\overline{)v=1.26\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.}$$

**Question b)**

**Data**

ω = 0.628 m

t = 2 min = 120 s

R = 2 m

φ_{0} = 0 rad (assume that the initial angle is 0 rad).

s_{0} = 0 m (assume that the initial distance traveled is 0 m).

**Resolution**

To calculate the angle covered:

$$\phi ={\phi}_{0}+\omega \xb7t\Rightarrow \phantom{\rule{0ex}{0ex}}\phi =0rad+0.628\raisebox{1ex}{$rad$}\!\left/ \!\raisebox{-1ex}{$\overline{)s}$}\right.\xb7120\overline{)s}\Rightarrow \phantom{\rule{0ex}{0ex}}\overline{)\phi =75.36rad}$$

and the distance traveled:

$$s=\phi \xb7R\Rightarrow \overline{)s=75.36rad\xb72m=150.72m}$$

**Question c)**

Since we are dealing with a u.c.m. the values of the accelerations that we can calculate in this type of motion are:

$$\overline{)\alpha =0{\displaystyle \raisebox{1ex}{$rad$}\!\left/ \!\raisebox{-1ex}{${s}^{2}$}\right.}}\vdots \overline{){a}_{n}=\frac{{v}^{2}}{R}={\omega}^{2}\xb7R=0.7887m/{s}^{2}}\vdots \overline{){a}_{t}=0{\displaystyle \raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{${s}^{2}$}\right.}}}$$