1 ) A Particle is moving in a vertical circle . The tensions in string when passing through two positions at angle 30° and 60° . Then T1 and T2 will be : -

1 )  A Particle is moving in a vertical circle . The tensions in string when passing  through two positions at angle 30° and 60° .  Then T1 and T2 will be : -

A ) T1 > T2 ,                                            B ) T1 < T2 ,

C ) T1 = T2 ,                                            D ) None of these ,

Correct Answer : -    A ) T1 > T2 ,

Solution : -

Formula : -

T = ( mv²/r² )  +  mgcos ( theta ) .

So angle = 30° ,

T1 = ( mv²/r² )  +  mgcos ( 30 ) ,

T1 = ( mv²/r² ) +  mg ( √3/2 ) .

So if angle = 60°  ,

T2 = ( mv²/r² )  +  mgcos ( 60° ) ,

T2 = (mv²/r²)  +  mg ( 1/2 ) .

So , T1 > T2  ,  Answer .

 

2 )  An electrical motor is required to haul a cage of mass 400 kg up a mine shaft through a vertical height of 1200 m in 2 minutes . What will be electrical power required , if the overall efficiency is 80 % ?

A ) 16  k W ,                                            B ) 32  k W ,

C ) 64  k W ,                                            D ) None of these ,

Correct Answer : -  B ) 32  k W ,

Solution : -

Given data : -

m = 400 kg ,

h = 1200 m ,

t = 2 mins = 120 seconds ,

power input = ?

Formula : -

Power output = w / t = f.d / t = mgh / t

Power output = ( 400 x 10 x 1200 ) / 120 ,

Power output = 40000 W .

So , we have efficiency %age formula : -

efficiency %  =  power output / power input .

power input = efficiency %  x  power output .

power input = 80/100 x 40000 = 32000 W = 32  k W  ,  Answer .