Question

An ant sits on a cd at a distance of 17 cm from the centre. If it sits there for 42 seconds, it travels a total distance of 913 cm. a. What angle has the ant turned through? (in radians) b. What speed has the ant been traveling at? (in cm/sec) c. What angular velocity has the ant been spinning at? (in radians/sec) d. What rpm is the cd turning at?

Answer 1

a) 53.7 rad

b) 21.7 cm/s

c) 1.28 rad/s

d) 12.2 rpm

Explanation:

a)

The ant sits on the cd at a distance of

r = 17 cm

from the centre: this is therefore the radius of the circle covered by the ant.

Therefore, we can find the length of the circumference of the circle:

$c=2\pi r=2\pi(17)=106.8 cm$

A full circle corresponds to an angle of

$\theta = 2\pi$ rad

In this problem instead, the distance covered by the ant along the circle is

d = 913 cm

Therefore, the angle corresponding to this distance can be found by using the rule of three:

$\frac{\theta}{c}=\frac{\theta'}{d}$

And so we find:

$\theta'=\theta \frac{d}{c}=(2\pi)\frac{913}{106.8}=53.7 rad$

b)

The speed of the ant can be calculated by using

$v=\frac{d}{t}$

where

d is the distance travelled by the ant

t is the time taken to cover that distance

v is the speed

In this problem:

d = 913 cm is the distance covered by the ant

t = 42 s is the time elapsed

Therefore, the speed of the ant is:

$v=\frac{913}{42}=21.7 cm/s$

c)

The angular velocity of an object in rotation is the rate of change of the angular displacement.

The angular velocity is related to the linear speed by

$\omega=\frac{v}{r}$

where

$\omega$ is the angular velocity

v is the linear speed

r is the radius of the circle

In this problem:

v = 21.7 cm/s is the speed of the ant

r = 17 cm is the radius of the circle

So, the angular velocity of the ant is:

$\omega=\frac{21.7}{17}=1.28 rad/s$

d)

The rpm means "revolutions per minute", and it is another units used to express the angular velocity.

In order to convert the angular velocity from radians per second to rmp, we must keep in mind that:

1 revolution = $2\pi$ radians

1 minute = 60 seconds

The angular velocity of the ant in this problem is

$\omega=1.28 rad/s$

Therefore, in order to convert to rpm, we apply the two conversion factors above, and so we get:

$\omega = 1.28 \frac{rad}{s}\cdot \frac{60 s/min}{2\pi rad/rev}=12.2 rev/min$

So, the cd is turning at 12.2 rpm.