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A particle of mass 100 g, is made to des...

A particle of mass 100 g, is made to describe a vertical circle of radius 1 m. Its instantaneous speed is `1ms^(-1)` when the string makes an sngle of `30^(@)` with the vertical Find the tension in the string at this position. Can the particle complete its circular path? `(g=10ms^(-2))`

Text Solution

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Hint : Tension in the string when it makes an angle `theta` with the vertical is
`T=(mv^(2))/(r )+mg cos theta`
Let speed at the lowest point be `v_(1)`
`v_(1)^(2)+v^(2)+2gr(1-cos theta)`
`(v_(1))_("min")=sqrt(5gr)" check if"v_(1) lt (v_(1))_("min")`
If yes then it would not compete its vertical circular path.
The tension in the string, when it makes an angle `theta` with the vertical is
`T=(mv^(2))/(r )+mg cos theta=(0.1xx1^(2))/(1)+0.1xx10xx0.866=0.966N`
Let the speed at the lowest point be `v_(1)`
`v_(1)^(2)=v^(2)+2gr(1-cos theta)`
`=1^(2)+2xx10xx1xx(1-0.866)`
`=1+20xx0.134`
`=3.68`
`v_(1)=sqrt(3.98)=1.91ms^(-1)`
`(v_(1))_("min")=sqrt(5gr)=sqrt(5xx10xx1)=7.07ms^(-1)`
`v_(1) lt (v_(1))_("min") rArr" the particle would not be able to complete its circular path."`
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