Final answer:
To find the cable tension needed to stop the elevator, use the work-energy principle to set the work done by tension equal to the change in kinetic energy. Factor in the effects of gravity and solve for tension to find that the required tension exceeds the elevator's weight alone.
Step-by-step explanation:
To determine the tension in the elevator cable, we must calculate the force needed to stop the elevator over the distance given, considering the elevator's mass and the acceleration due to gravity.
First, we can use the work-energy principle that states that the work done on the elevator by the tension in the cable is equal to the change in kinetic energy.
The kinetic energy (KE) of the elevator at 3.8 m/s is given by KE = (1/2)mv2, where m is the mass and v is the velocity.
Since the elevator is coming to a stop, its final kinetic energy is zero, so the work done by the tension (W = F * d, where F is the force and d is the distance) is equal to the negative of the initial kinetic energy.
Thus,
W = -KE = -(1/2) * m * v2.
Now, considering the work done by the tension and gravity,
we use W = Ftension * d - m * g * d, where g is the acceleration due to gravity (9.8 m/s2).
Solving for the tension (Ftension) gives: Ftension = (m * g + (1/2) * m * v2)/d
After plugging in the given values (m = 1320 kg, v = 3.8 m/s, g = 9.8 m/s2, d = 3.3 m), we get Ftension.
It should be noted that the weight of the elevator (mg) is part of the tension when the elevator is stationary, and additional tension is required to stop it from its downward motion.
Therefore, the tension required would be more than just the weight of the elevator.