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Answer :
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:
[tex]c=2\pi r=2\pi(17)=106.8 cm[/tex]
A full circle corresponds to an angle of
[tex]\theta = 2\pi[/tex] 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:
[tex]\frac{\theta}{c}=\frac{\theta'}{d}[/tex]
And so we find:
[tex]\theta'=\theta \frac{d}{c}=(2\pi)\frac{913}{106.8}=53.7 rad[/tex]
b)
The speed of the ant can be calculated by using
[tex]v=\frac{d}{t}[/tex]
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:
[tex]v=\frac{913}{42}=21.7 cm/s[/tex]
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
[tex]\omega=\frac{v}{r}[/tex]
where
[tex]\omega[/tex] 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:
[tex]\omega=\frac{21.7}{17}=1.28 rad/s[/tex]
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 = [tex]2\pi[/tex] radians
1 minute = 60 seconds
The angular velocity of the ant in this problem is
[tex]\omega=1.28 rad/s[/tex]
Therefore, in order to convert to rpm, we apply the two conversion factors above, and so we get:
[tex]\omega = 1.28 \frac{rad}{s}\cdot \frac{60 s/min}{2\pi rad/rev}=12.2 rev/min[/tex]
So, the cd is turning at 12.2 rpm.
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Rewritten by : Barada
a) 53.7 rad
b) 21.7 cm/s
c) 1.28 rad/s
d) 12.2 rpm
Given:
Radius, r= 17 cm
Total distance, d=913 cm
a)
We need to find the length of the circumference of the circle:
[tex]c=2\pi r\\\\c=2*3.14*17\\\\c=106.8cm[/tex]
In order to find an angle has the ant turned through we will use:
[tex]\theta=2\pi *rad[/tex]
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:
[tex]\frac{\theta}{c} =\frac{\theta^'}{d} \\\\\theta^'=\frac{\theta*d}{c}\\\\\theta^'=2\pi\frac{913}{106.8} \\ \\\theta^'=53.7rad[/tex]
b)
The speed of the ant can be calculated by using:
[tex]\text{speed}=\frac{\text{distance}}{\text{time}} \\\\[/tex]
where
d is the distance travelled by the ant
t is the time taken to cover that distance
v is the speed
[tex]\text{speed}=\frac{\text{distance}}{\text{time}} \\\\\text{speed}=\frac{913}{42} \\\\\text{speed}=21.7cm/s[/tex]
Therefore, the speed of the ant is: 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.
[tex]\omega=\frac{v}{r}[/tex]
where
[tex]\omega[/tex] is the angular velocity
v is the linear speed
r is the radius of the circle
Given: 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:
[tex]\omega=\frac{v}{r}\\\\\omega=\frac{21.7}{17} \\\\\omega=1.28 rad/s[/tex]
d)
The rpm means "revolutions per minute", and it is another units used to express the angular velocity.
Unit conversions:
1 revolution = radians
1 minute = 60 seconds
The angular velocity of the ant in this problem is
[tex]\omega=1.28\frac{rad}{s} *\frac{60s/min}{2\pi rad/rev} =12.2 rev/min[/tex]
So, the cd is turning at 12.2 rpm.
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