A solid sphere of radius R is placed at a height of 30 cm on a15 degree slope. I

Question

A solid sphere of radius R is placed at a height of 30 cm on a15 degree slope. It is released and rolls, without slipping, to thebottom. a) From what height should a circular hoop of radius R bereleased on the same slope in order to equal the sphere's speed atthe bottom? b) Can a circular hoop of different diameter be released froma height of 30 cm and match the sphere's speed at the bottom? If,so what is the diameter? If not, why not? A solid sphere of radius R is placed at a height of 30 cm on a15 degree slope. It is released and rolls, without slipping, to thebottom. a) From what height should a circular hoop of radius R bereleased on the same slope in order to equal the sphere's speed atthe bottom? b) Can a circular hoop of different diameter be released froma height of 30 cm and match the sphere's speed at the bottom? If,so what is the diameter? If not, why not?

Explanation / Answer

For a solid sphere the     m g h = (1/2)I2 +(1/2)mV2   Here I = (2/5)mR2   and V= R So          m g h =(1/2)(2/5)m R2 ( V2 /R2 ) +(1/2)mR2           m g h = (7 /10) mV2            V = [10 g h/ 7 ] ------------------------------------------------------------------------------------------------------------- For circular hoop    m g h = (1/2)I2 +(1/2)mV2      Here I = mR2   and V= R So           m g h =(1/2)m R2 ( V2 /R2 ) +(1/2)mR2             m g h =mV2             h =V2 / g             = [10 g h / 7 ] / g               =10h / 7              = 10x 30cm/ 7               = 42.857m --------------------------------------------------------------------------------------------------------------------- The speed of the hoop is independent of the radius ofthe hoop.    Here I = mR2   and V= R So           m g h =(1/2)m R2 ( V2 /R2 ) +(1/2)mR2             m g h =mV2             h =V2 / g             = [10 g h / 7 ] / g               =10h / 7              = 10x 30cm/ 7               = 42.857m --------------------------------------------------------------------------------------------------------------------- The speed of the hoop is independent of the radius ofthe hoop.   

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