Units and Measurement

Question

A solid cylinder rolls up an inclined plane of angle of inclination 30°. At the bottom of the inclined plane the centre of mass of the cylinder has a speed of 5 m/s.

(a) How far will the cylinder go up the plane?

(b) How long will it take to return to the bottom?

Solution

Initial velocity of the solid cylinder, v = 5 m/s
Angle of inclination, θ = 30°
Let the cylinder go up the plane upto a height h.

Using the relation,

1 half space mv squared space plus space 1 half space straight I space straight omega squared space equals space mgh

1 half space mv squared space plus space 1 half open parentheses 1 half space mr squared close parentheses straight omega squared space equals space mgh space 3 over 4 mv squared space equals space mgh space straight h space equals 3 fraction numerator straight v squared over denominator 4 straight g end fraction space equals space 3 space straight x space fraction numerator 52 over denominator 4 space straight x space 9.8 end fraction space space space equals space 1.913 space straight m space

If s is the distance up the inclined plane, then as
sin θ =

open parentheses straight h over straight s close parentheses

s =

open parentheses fraction numerator straight h over denominator sin space straight theta end fraction close parentheses

open parentheses fraction numerator 1.913 over denominator sin space 30 to the power of straight o end fraction close parentheses space equals space 3.826 space straight m

Therefore, time taken to return to the bottom of the inclined plane,

straight t space equals space square root of fraction numerator 2 space straight s space open parentheses 1 space plus space bevelled straight k squared over straight r squared close parentheses over denominator straight g space sin space straight theta end fraction end root space space space equals space fraction numerator 2 space straight x space 3.826 space open parentheses 1 space plus space bevelled 1 half close parentheses over denominator 9.8 space sin space 30 to the power of straight o end fraction space space space equals space 1.53 space sec

1.53 sec is required for the balloon to return to the bottom.