In one type of elevator, the car (in which people ride) is supported from above
ID: 1872906 • Letter: I
Question
In one type of elevator, the car (in which people ride) is supported from above by a group of cables that hang over a big pulley at the top of the elevator shaft. The other ends of the cables are attached to a counterweight which weighs about the same as the elevator car plus a few passengers. There's a nice animated figure here: http://science.howstuffworks.com/transport/engines-equipment/elevator3.htm. The elevator motor turns the pulley to move the car up and counterweight down, or vice versa. Let's think about a simplified elevator, with a single cable (and without the safety mechanisms that real elevators have) and consider a particular failure mode: suppose the motor shaft breaks, so the pulley now rotates freely and the elevator car and counterweight will move up or down depending on the net forces on them, while still connected by the cable. Use me to represent the mass of the elevator car including the passengers inside, and let mc be the mass of the counterweight. Let's go step by step (a) Choose a suitable coordinate system and identify all of the forces on the elevator car, saying whether each one is positive or negative. Do the same for all of the forces on the counterweight. (b) Using Newton's second law, what is the acceleration of the elevator car, expressed symbolically in terms of things like me and the tension in the cable? (c) Using similar reasoning, what is the acceleration of the counterweight, expressed symbolically in terms ofm. , etc.? (d) Now use what you know about the relationship between the accelerations of the elevator car and the counterweight to combine your answers from parts (b) and (c), and from that do the math to determine the tension in the cable and the acceleration of the elevator car So far you've done the calculation symbolically; now it's time for some interpretation. (e) What would the acceleration be if there were no counterweight, i.e. if mc-0? Does that make sense? (f) What would the acceleration and the cable tension be if the elevator car (including passengers) and the counterweight have equal masses, i.e. if me - mc ? Does that make sense? (g) Suppose the elevator car and passengers have a total mass of 650 kg, and the counterweight has a mass of 500 kg. If this type of failure occurred, how would the elevator car move? Would you be alarmed if you were riding in the car at the time?Explanation / Answer
given
mass of elevator car = me
mass of counter weight = mc
a. forces on elevator car : tension due to string, force of gravity
forces on counter weight : tension due to string, force of gravity
the force of gravity always acts downwards, and the tension acts away from the body
the motion of one of the bodies can be considered downwards giv ing us sign of the forces involved
b. let acceleration of the system be a, and let the counter weight be moving downwards
then
from newtons laws, if tension in the cable is T
T - me*g = me*a
c. mcg - T = mc*a
d. from both the equations
a = g(mc - me)/(mc + mg)
T = me(g + g(mc - me)/(mc + me))
T = g*me(mc + me + mc - me)/(mc + me)
T = 2me*mc*g/(mc + me)
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