Ok, you go up but how far?
Take a solid cylinder of radius R , roll it with angular velocity ω 0 and carefully keep it on an inclined plane having inclination θ , and coefficient of kinetic friction μ k , such that μ k > tan θ . Find the maximum height attained by the cylinder above the initial position in m m to the nearest integer
Details and Assumptions
- ω 0 = 5 0 r a d s − 1
- R = 2 5 c m
- μ k = 3 1
- tan θ = 4 1
- g = 9 . 8 m / s 2
The answer is 443.
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2 solutions
I did like this, 1)Conserved angular momentum along the plane 2)Applied work energy theorem to find distance travelled during the change to pure rolling condition 3)took the sine of distance to get height 4) Conserved energy to find next increment in height 5) then added them to get ans as2467 Please guide me where I went wrong.
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How could you use conservation of angular momentum? In this case obviously angular momentum is not conserved.
Can you write the detailed solution? I am not getting the correct answer
That is exactly what I did,but my answer is wrong , can you write the expression you got for the time ?
very nice problem albeit lengthy.
height attained to achieve pure rolling is 0.295m and height attained till momentary rest is 0.1475m. Summing them up gives 0.443m
First write force equation : μ m g c o s θ − m g s i n θ = m d v / d t
and then torque equation: μ m g c o s θ × R = I ( − d ω / d t )
Integrating both expressions;
we get velocity at the time of rolling, v= T ( μ g c o s θ − g s i n θ )
and angular velocity at the time of rolling, ω = ω o − 2 μ g c o s θ T / R
where T is the time taken to attain pure rolling. And also in case of pure rolling
v = R ω
If you notice that from the expression of 'v', we can also get the distance travelled by cylinder (s) as a function of time
s = ( μ g c o s θ − g s i n θ ) t 2 / 2
The height gained is therefore
h 1 = s . s i n θ
Solve all the equations to obtain 'h1'. Then, apply conservation of energy to find 'h2'.
take out time required to achieve pure rolling...................................................calculate the distance covered.............................................then use conservation of energy......................add the 2 heights to get the final answer