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Centripetal Acceleration

In the figure...

Question

In the figure shown below, a mass $m_{1}$ is placed on a horizontal frictionless table, and connected to a mass $m_{2}$ by a string of negligible mass. Mass $m_{2}$ remains stationary and $m_{1}$ moves in a circular path of radius $^{'} r^{'}$ with an angular speed $^{'} ω^{'} r a d / s$ . If mass $m_{2}$ is increased by placing an additional load on it, which of the following statements is/are true?

Speed of mass

m_{1}

remains same

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Mass

m_{1}

spirals inward by gaining speed

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Gradually increasing tension (

T

) in the string produces tangential acceleration of mass

m_{1}

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Tension in the string produces inward radial acceleration of mass

m_{1}

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Solution

The correct options are
B Mass $m_{1}$ spirals inward by gaining speed
C Gradually increasing tension ( $T$ ) in the string produces tangential acceleration of mass $m_{1}$
D Tension in the string produces inward radial acceleration of mass $m_{1}$
For mass $m_{2}$ , we have $T = m_{2} g$
So, when mass $m_{2}$ is increased, it causes an increase in the tension force $T$ .
Since tension provides necessary centripetal force, it will lead to increased radial acceleration, due to which the mass $m_{1}$ moves towards the centre i.e option (d) is correct.

From equation of circular motion
$T = m r ω^{2}$ i.e $ω \propto \frac{1}{\sqrt{r}}$
As $r$ decreases, $ω$ increases, thereby increasing tangential acceleration $(a_{t})$ i.e option (c) is correct.

$∴ m_{1}$ will spiral inwards by gaining speed. Option (b) is correct.

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