Final answer:
To find the initial speed of the descending elevator, we calculate the net force by subtracting the weight from the tension, then apply the work-energy principle to find the deceleration of the elevator and, finally, use the calculated deceleration to derive the initial velocity.
Step-by-step explanation:
The question asks for the initial speed of a descending elevator which is decelerated to rest by a tension force in the cable. We know the elevator's mass (420 kg), the tension in the cable (5519 N), the distance over which it decelerates (4 m), and the acceleration due to gravity (9.8 m/s²).
First, we must determine the deceleration of the elevator using Newton's second law. The net force acting on the elevator is the difference between the tension in the cable and the gravitational force (weight of the elevator). This net force is responsible for decelerating the elevator:
Net force (Fnet) = Tension (T) - Weight (W) = ma
Where:
m is the mass of the elevator
a is the acceleration (deceleration in this case)
T is the tension in the cable
W is the weight of the elevator (m ⋅ g)
Let's calculate the deceleration (a):
a = (T - m ⋅ g) / m
Work-energy principle states that the work done by the net force on the elevator equals the change in kinetic energy.
The work done (W) by the net force is the product of the net force and the distance (d) over which it acts:
W = Fnet ⋅ d
The change in kinetic energy (ΔKE) is:
ΔKE = 1/2 m vf² - 1/2 m vi²
Since the final velocity (vf) is 0 m/s (the elevator comes to a rest), the change in kinetic energy is simply:
ΔKE = -1/2 m vi²
By equating the work done to the change in kinetic energy, we can solve for the initial velocity (vi):
vi = √(2 ⋅ Fnet ⋅ d / m)
After calculating the net force and then the initial velocity using the above formulas, we can find the speed (vi) of the elevator at the beginning of the 4m descent.