Speed of a fan

**m_i_k_o** · October 2nd 2009, 05:03 PM

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?

**skeeter** · October 2nd 2009, 05:38 PM

Originally Posted by m_i_k_o

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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