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A solid cylinder rolls up an inclined plane of angle of inclin

Question

A solid cylinder rolls up an inclined plane of angle of inclination $3 0^{o}$ . At the bottom of the inclined plane the center 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?

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A solid cylinder rolls up an inclined plane of angle of inclination

3 0^{o}

. At the bottom of the inclined plane the center 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?

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Text solutionVerified

(a) Let the cylinder roll up to height

h

From conservation of energy between the initial and final states,

\frac{1}{2} m v^{2} + \frac{1}{2} I ω^{2} = m g h

and,

I = \frac{1}{2} m r^{2}

⟹ \frac{1}{2} m v^{2} + \frac{1}{2} (\frac{1}{2} m r^{2}) ω^{2} = m g h

Also for rolling,

v = r ω

Thus

h = \frac{3 v ^{2}}{4 g} = 1.913 m

If s is distance it goes up the plane,

s in θ = \frac{h}{s}

Thus

s = \frac{h}{s in θ} = 3.826 m

(b) Time taken to return the bottom=

\frac{2 s ( 1 + \frac{K ^{2}}{r ^{2}} )}{g sin θ} = \frac{2 \times 3.826 \times ( 1 + \frac{1}{2} )}{9.8 \times \frac{1}{2}} = 1.53 s

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Question Text	A solid cylinder rolls up an inclined plane of angle of inclination $3 0^{o}$ . At the bottom of the inclined plane the center 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?
Answer Type	Text solution:1
Upvotes	150