A solid cylinder of radius is set into rotation about its axis with angular speed then lowered with its lateral surface onto a horizontal plane and released. If the coefficient of friction is , find after how much time (in seconds) will the cylinder start pure rolling.
If the time is given by , find .
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We have the following equations
F ( ext. force ) = f k = μ m g = m a ⟹ a = μ g
Γ ( ext. torque ) = f k R = I α ⟹ μ m g R = 2 m R 2 α ⟹ α = R 2 μ g
Now using the equations
ω = ω 0 + α t
and
v = v 0 + a t
we get
ω = ω 0 − R 2 μ g t ⋯ ( 1 ) ( negative sign because of opposite sense of α and ω 0 )
v = μ g t ⋯ ( 2 )
At the instant when pure rolling starts, we will have v = R ω . Thus equation ( 2 ) becomes
R ω = μ g t ⋯ ( 3 )
Solving ( 1 ) and ( 3 ) , we get
t = 3 μ g R ω 0