Question

In a friction winder, the skip accelerates to a steady speed in 15 s. The torque vs. time diagram is shown. Find the deceleration time (in seconds, rounded to one decimal place).

Hint:

Torque difference above or below running torque determines angular acceleration and deceleration.

Answer 1

Correct Answer: 10

The torque–time diagram has three levels: - Acceleration torque = 70 kNm - Running torque = 50 kNm - Deceleration torque = 20 kNm Since torque is proportional to angular acceleration: \[ \frac{\alpha_{\text{acc}}}{\alpha_{\text{dec}}} = \frac{T_{\text{acc}} - T_{\text{run}}}{T_{\text{run}} - T_{\text{dec}}} \] Acceleration torque available: \[ T_{\text{acc}} - T_{\text{run}} = 70 - 50 = 20\ \text{kNm} \] Deceleration torque available: \[ T_{\text{run}} - T_{\text{dec}} = 50 - 20 = 30\ \text{kNm} \] Thus: \[ \frac{\alpha_{\text{acc}}}{\alpha_{\text{dec}}} = \frac{20}{30} = \frac{2}{3} \] Since: \[ t_{\text{acc}} = 15\ \text{s} \] And: \[ t \propto \frac{1}{\alpha} \] \[ \frac{t_{\text{dec}}}{t_{\text{acc}}} = \frac{\alpha_{\text{acc}}}{\alpha_{\text{dec}}} = \frac{2}{3} \] \[ t_{\text{dec}} = 15 \times \frac{2}{3} = 10\ \text{s} \] Thus: \[ \boxed{10.0\ \text{s}} \]