One elevator arrangement includes the passenger car, a counterweight, and two large pulleys, as shown in (Figure 1). Each pulley has a radius of 1.3 m and a moment of inertia of 370 kg?m2. The top pulley is driven by a motor. The elevator car plus passengers has a mass of 3200 kg, and the counterweight has a mass of 2600 kg.
If the motor is to accelerate the elevator car upward at 1.5
m/s2, how much torque must it generate?
Express your answer to two significant figures and include appropriate units.
Value and Units
To accelerate the elevator car upward at 1.5 m/s², we need a rope tension
T2 = 3200kg * (9.8 + 1.5)m/s² = 36 160 N
and to have the counterweight accelerate downward at 1.5 m/s²,
T1 = 2600kg * (9.8 - 1.5)m/s² = 21 580 N
When the linear acceleration is 1.5 m/s², the angular acceleration of the pulleys is
? = a / r = 1.5m/s² / 1.3m = 1.154 rad/s²
Each pulley has a moment of inertia
I = ½mr² = ½ * 370kg * (1.3m)² = 312.65 kg·m²
and there are two of them.
So, we'd say that the total required torque is
? = 2*I*? + (T2 - T1)*r
? = 2 * 312.65 kg·m² * 1.154rad/s² + (36160 - 21580)N * 1.3m
? = 19675.6 N·m
= 2.0 x 104 N·m (2 sf)
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