Respuesta :
The speed of the elevator after covering after the point of its contact with the spring will be [tex]\boxed{3.65\,{{\text{m}} \mathord{\left/{\vphantom {{\text{m}} {\text{s}}}} \right.\kern-\nulldelimiterspace} {\text{s}}}}[/tex].
Further Explanation:
Given:
The mass of the elevator is [tex]2000\,{\text{Kg}[/tex].
The speed of the elevator after the cable breaks is [tex]4\,{{\text{m}} \mathord{\left/{\vphantom {{\text{m}} {\text{s}}}}\right.\kern-\nulldelimiterspace} {\text{s}}}[/tex].
The friction force by the safety clamp is [tex]17000\,{\text{N}[/tex].
Concept:
As the cable of the elevator breaks, the elevator starts to fall with some velocity and the friction provided by the cushion spring will start to decrease the speed of the elevator.
The total amount of kinetic energy of the elevator as it starts moving is:
[tex]\begin{aligned}{K_i}&= \frac{1}{2}m{v^2}\\&=\frac{1}{2}\times 2000 \times {\left( 4 \right)^2}\\&= 16000\,{\text{J}}\\\end{aligned}[/tex]
The net force acting on the elevator during its drag of over the safety clamp is:
[tex]\begin{aligned}{F_{net}} &= {F_g} - f\\&= mg - f\\&= \left( {2000 \times 9.8} \right) - 17000\\&= 2600\,{\text{N}}\\\end{aligned}[/tex]
Finally when the elevator covers distance, the elevator has some kinetic energy due to its speed and rest of the energy is exhausted in the work done by the net force in covering the next over the spring.
The work done by the force in covering [tex]1\,{\text{m}}[/tex] is:
[tex]\begin{aligned}{W_i}&= {F_{net}} \times d\\&= 2600 \times 1\\&= 2600\,{\text{J}}\\\end{aligned}[/tex]
Apply the conservation of energy for the elevator.
[tex]\begin{aligned}{K_i}&= {K_f} + {W_i}\\16000&= \frac{1}{2}mv_f^2 + 2600\\{v_f}&=\sqrt{\frac{{2 \times 13400}}{{2000}}}\\ &=3.66\,{{\text{m}}\mathord{\left/{\vphantom {{\text{m}} {\text{s}}}}\right.\kern\nulldelimiterspace}{\text{s}}}\\\end{aligned}[/tex]
Thus, the speed of the elevator after covering after the point of its contact with the spring will be [tex]\boxed{3.65\,{{\text{m}} \mathord{\left/{\vphantom {{\text{m}} {\text{s}}}}\right.\kern-\nulldelimiterspace} {\text{s}}}}[/tex].
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Answer Details:
Grade: College
Subject: Physics
Chapter: Conservation of Energy
Keywords: Elevator, worst-case, design scenario, broken cables, contacts a cushioning spring, stop the elevator, conservation of energy, initial kinetic energy, work done.
