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Question
Starting from rest, an object rolls down along an incline that rises by 3 in every 5 (along with it). The object gains a speed of `sqrt 10` m/s as it travels a distance of `5/3` m along the incline. What can be the possible shape/s of the object?
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Solution
Given: Incline that rises by 3 in every 5 is sin θ = `3/5`
The object gains a speed of v = `sqrt10` m/s
It travels a distance along the incline s = `5/3` m
To find: The shape of the possible object, i.e., to find out the ratio of `K^2/R^2` which will determine the possible rolling object.
Solution: We have, `sin theta = 3/5`
And Linear distance travelled along the plane is s = `h/sin theta`
Hence,
h = s sin θ = `5/3 xx 3/5` = 1
The velocity of rolling body is given by,
v = `sqrt((2gh)/(1 + K^2/R^2))`
Comparing we get,
`sqrt 10 = sqrt((2gh)/(1 + K^2/R^2))`
10 = `(2 xx 10 xx 1)/(1 + (K^2)/R^2)`
`(1 + (K^2)/R^2)` = 2
`(K^2)/R^2` = 1
Hence object must be Ring or hollow cylinder.
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