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Math Help - Speed of a fan

  1. #1
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    Speed of a fan

    A large mine ventilation fan is rotating at a speed of 120 RPM when the electricity is turned off. The fan then decelerates due to friction at a rate of -0.05ω rad/s2, where ω is the angular velocity in rad/s.

    How long does it take for the fan speed to come down to N RPM?

    N[rpm] = 6;
    Could someone please show me how to solve this?
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  2. #2
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    Quote Originally Posted by m_i_k_o View Post
    A large mine ventilation fan is rotating at a speed of 120 RPM when the electricity is turned off. The fan then decelerates due to friction at a rate of -0.05ω rad/s2, where ω is the angular velocity in rad/s.

    How long does it take for the fan speed to come down to N RPM?

    N[rpm] = 6;
    Could someone please show me how to solve this?
    first, you need to convert \omega_o = 120 RPM and \omega_f = 6 RPM to rad/sec

    \alpha = \frac{d\omega}{dt} = -0.05\omega

    \frac{d \omega}{\omega} = -0.05 \, dt

    \ln{\omega} = -0.05t + C

    \omega = e^{-0.05t + C} = e^C \cdot e^{-0.05t} = Ae^{-0.05t}

    sub in the initial condition ...

    \omega_o  = Ae^{-0.05 \cdot (0)} = A \cdot 1 ... A = \omega_o

    \omega = \omega_o e^{-0.05t}

    sub in \omega_o and \omega_f in rad/sec and solve for t
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