A 300kg elevator has passengers with unknown mass. From rest, the elevator accelerates upward at 1.2 m/s². The cable supporting the elevator exerts on upward force of 8000 N. What is the mass of the passengers that the elevator carries?
Answer (1)
[tex]\begin{gathered}\begin{gathered}{\underline{\huge \mathbb{A} {\large \mathrm {NSWER : }}}} \\\end{gathered}\end{gathered}[/tex] To find the mass of the passengers in the elevator, we can use Newton's second law and the information provided about the forces acting on the elevator. Identify the known values Mass of the elevator, [tex]m_e = 300 \text{ kg}[/tex] Acceleration of the elevator, [tex]a = 1.2 \text{ m/s}^2[/tex] Upward force exerted by the cable, [tex]F_{\text{cable}} = 8000 \text{ N}[/tex] Calculate the total mass being lifted Let [tex]m_p[/tex] be the mass of the passengers. The total mass being lifted is: [tex]m_{\text{total}} = m_e + m_p[/tex] Apply Newton's second law According to Newton's second law, the net force [tex]F_{\text{net}}[/tex] acting on the elevator can be expressed as: [tex]F_{\text{net}} = m_{\text{total}} \cdot a[/tex] We also know that the net force is the difference between the upward force exerted by the cable and the weight of the total mass: [tex]F_{\text{net}} = F_{\text{cable}} - m_{\text{total}} \cdot g[/tex] Where g is the acceleration due to gravity ([tex]g \approx 9.81 \text{ m/s}^2[/tex]). Set up the equation Equating both expressions for the net force gives us: [tex]m_{\text{total}} \cdot a = F_{\text{cable}} - m_{\text{total}} \cdot g[/tex] Substitute [tex]m_{\text{total}}[/tex] Substituting [tex]m_{\text{total}}[/tex] into the equation, we have: [tex](m_e + m_p) \cdot a = F_{\text{cable}} - (m_e + m_p) \cdot g[/tex] Rearranging the equation Rearranging the equation: [tex](m_e + m_p) \cdot a + (m_e + m_p) \cdot g = F_{\text{cable}}[/tex] Factoring out ([tex]m_e + m_p[/tex]): [tex](m_e + m_p) \cdot (a + g) = F_{\text{cable}}[/tex] Solve for [tex]m_{\text{total}}[/tex] Now we can solve for [tex]m_{\text{total}}[/tex]: [tex]m_e + m_p = \frac{F_{\text{cable}}}{a + g}[/tex] Plug in the values Substituting the values we have: [tex]m_e + m_p = \frac{8000 \text{ N}}{1.2 \text{ m/s}^2 + 9.81 \text{ m/s}^2}[/tex] Calculating the denominator: [tex]a + g = 1.2 + 9.81 = 11.01 \text{ m/s}^2[/tex] Now substituting: [tex]m_e + m_p = \frac{8000}{11.01} \approx 726.44 \text{ kg}[/tex] Solve for the mass of the passengers Since we know [tex]m_e = 300 \text{ kg}[/tex], we can find [tex]m_p[/tex]: [tex]m_p = 726.44 \text{ kg} - 300 \text{ kg} \approx 426.44 \text{ kg}[/tex] Conclusion Therefore, the mass of the passengers that the elevator carries is approximately 426.44 kg. [tex]\sf\color{green}{⊱⋅ ────────────────────── ⋅⊰}[/tex][tex]\begin{gathered} \boxed{\begin{array}{cc} \sf \footnotesize \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: ···\: ʚ\: \: \: \: \: \: Hope\:it\:helps \: \: \: \: \: ɞ \:··· \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \sf\footnotesize\:\# CarryOnLearning \\ \sf\footnotesize ૮₍´˶ \: • \: . \: • \: ⑅ ₎ა \: \leadsto \footnotesize\sf\color{purple} \underline{Study\:Well!}\end{array}}\end{gathered}[/tex][tex]\sf\color{green}{⊱⋅ ────────────────────── ⋅⊰}[/tex][tex]\large\qquad\qquad\qquad\tt MARCH/14/2025 [/tex]
