Question

Exumple 4.10 An insect trapped in a
circular groove of radius 12 cm moves along
the groove steadily and completes 7
1 evolutions in 100 s. (a) What is the
angular speed, and the linear speed of the
motion? (b) Is the acceleration vector a
constant vector ? What is its magnitude ?​

Answer 1

$\huge\frak{AnswEr :}$

Given that, an insect is trapped in a circular groove with radius 12 cm which moves along the groove steadily & completes 7 revolutions in 100 sec.

That is,

• Radius of the groove = 12 cm
• No. of revolutions = 7
• Time taken to complete those revolutions = 100 sec.

We've to calculate the following values :

▪︎ What is the angular speed, and the linear speed of the motion?

▪︎ Is the acceleration vector a constant vector ?What is its magnitude ?

Here you go,

In order to calculate angular speed, linear speed we've certain formula's & by substituting the values we can obtain the required answer.

So,

• Frequency[Revolutions] of Insect = Total No. of revolutions completed in 1 second

That is,

$\longrightarrow\sf \dfrac{No. \: of \: revolutions \: }{ 100}$

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Substituting the values :

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$\longrightarrow\sf \dfrac{7}{100}$

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$\longrightarrow \large{ \boxed{\sf 0.07 \: s}}$

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I) We know that,

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$\longrightarrow\sf Angular \: Speed = 2 \pi n$

Substituting the values :

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$\longrightarrow\sf Angular \: Speed = 2 \times 0.07$

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$\longrightarrow\sf Angular \: Speed = 2 \times \dfrac{1}{2} \times 0.07$

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$\longrightarrow{ \blue{\sf Angular \: Speed = 0.44 \: rad /s }}$

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Also,

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$\longrightarrow\sf Linear \: Speed = \omega \times r$

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Substituting the values :

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$\longrightarrow\sf Linear \: Speed = 0.44 \times 0. 12$

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$\longrightarrow{ \green{\sf Linear \: Speed = 0.0528 m/s}}$

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II) Now we've to find that whether the acceleration is constant vector or not with it's magnitude.

According to the question, I guess that the direction of velocity is continuously changing, that is, it is not stable. Hence, the acceleration is not a constant vector.

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As the acceleration performed is Centripetal,

$\longrightarrow\sf Magnitude = \omega^2 \times r$

$\longrightarrow\sf Magnitude = 0.44^2 \times 0.12$

$\longrightarrow\sf \purple{Magnitude = 0.0232 \: rad/s^2}$