An elevator in a tall building is allowed to reach a maximum speed of 3.61 m/s going down. What must the tension be in the cable to stop this elevator over a duration of 3 seconds if the elevator has a mass of 1370 kg including occupants?
a = 3.61/3 up
Force down = m g = 1370 (9.81)
Force up = T
so
F up = m (a up)
T - 1370 (9,81) = 1370 (3.61/3)
To determine the tension in the cable required to stop the elevator over a duration of 3 seconds, we can use the equation of motion:
F = ma
Where:
F = Force (tension in the cable)
m = Mass of the elevator
a = Acceleration
To calculate the acceleration, we can use the equation:
a = Δv / Δt
Where:
Δv = Change in velocity (from 3.61 m/s to 0 m/s)
Δt = Change in time (3 seconds)
First, let's calculate the acceleration:
Δv = 0 m/s - (-3.61 m/s) = 3.61 m/s
Δt = 3 seconds
a = Δv / Δt
a = 3.61 m/s / 3 s
a ≈ 1.203 m/s²
Next, we can calculate the force (tension) in the cable:
F = ma
F = (1370 kg) * (1.203 m/s²)
F ≈ 1649.01 N
Therefore, the tension in the cable must be approximately 1649.01 N to stop the elevator over a duration of 3 seconds.
To determine the tension in the cable required to stop the elevator, we need to consider the concept of force and acceleration.
First, let's calculate the acceleration of the elevator, as acceleration is the rate of change of velocity. We can use the kinematic equation:
v = u + at
Where:
v = final velocity (0 m/s as the elevator needs to come to a stop)
u = initial velocity (3.61 m/s downwards)
a = acceleration
t = time taken to stop (3 seconds)
Plugging in the values, we can rearrange the equation to solve for acceleration:
0 = 3.61 m/s + a * 3 seconds
-3.61 m/s = 3a seconds
a = -1.2033 m/s^2
Note that the acceleration is negative as it is opposing the initial downward velocity.
Now that we have the acceleration, we can calculate the net force acting on the elevator. The net force is equal to the product of mass and acceleration:
F = m * a
Where:
F = net force (tension in the cable)
m = mass of the elevator (1370 kg)
a = acceleration (-1.2033 m/s^2)
Plugging in the values:
F = 1370 kg * -1.2033 m/s^2
F ≈ -1651.18 N
Since tension is a force, it must be positive. Therefore, we take the absolute value of the force:
Tension = |F| = |-1651.18 N|
Tension ≈ 1651.18 N
Therefore, the tension in the cable required to stop the elevator over a duration of 3 seconds is approximately 1651.18 Newtons.