Question

You are designing a simple elevator system for an old warehouse that is being converted to loft apartments. A 22,500-N elevator is to be accelerated upward by connecting it to a counterweight by means of a light (but strong!) cable passing over a solid uniform disk-shaped pulley. The cable does not slip where it is in contact with the surface of the pulley. There is no appreciable friction at the axle of the pulley, but its mass is 871 kg and it is 1.30 m in diameter.
A. What mass should the counterweight have so that it will accelerate the elevator upward through 6.75 m in the first3.00 s, starting from rest?
B. Under these conditions, what is the tension in the cable on the elevator side of the pulley?
C. Under these conditions, what is the tension in the cable on the counterweight side of the pulley?

Answer 1

Answer:

A) mass = 3121.58 kg

B) tension = 25940.37 N

C) tension = 25940.37 N (tension on both sides will be the same)

Explanation:

Weight of elevator = 22500 N

Distance = 6.75 m

Time = 3 sec

Since it started from rest, initial speed is zero.

Using Newton's equation of motion we have,

S = ut + 0.5at^2

S = distance covered = 6.75 m

t = time = 3 s

a = acceleration upwards

u = initial velocity = 0

Substituting values, we have,

6.75 = 0(3) + (0.5 x a x 3^2)

6.75 = 4.5a

a = 6.75/4.5 = 1.5 m/s^2 (acceleration of the elevator upwards)

Mass of the elevator = Weight/g

Where g = acceleration due to gravity 9.81 m/s

Mass = 22500/9.81 = 2293.58 kg

From the image below we solve from

T - 22500 = ma

T - 22500 = 2293.58 x 1.5

T - 22500 = 3440.37

T = 3440.37 + 22500 = 25940.37 N (this is the tension on the rope)

On the other side,

mg - T = ma

9.81m - 25940.37 = 1.5m

(9.81 - 1.5)m = 25940.37

8.31m = 25940.37

m = 3121.58 kg (mass of counter weight)

See image below