Source: Principles of Physics
Problem
A lamp of mass $m$ hangs from a cord in an elevator. The elevator experiences an upward acceleration of magnitude $a$, and the acceleration due to gravity is $g$.
- The elevator descends and decelerates, resulting in an upward acceleration of magnitude $a$. If the tension in the cord is $T$, what is the lamp's mass $m$?
- What is the cord's tension $T'$ when the elevator ascends with the same upward acceleration of magnitude $a$?
[Q1] $m = \frac{T}{g+a}$ [Q2] $T' = T$
Let the upward direction be positive. The forces on the lamp are the upward tension from the cord and the downward gravitational force $mg$.
Applying Newton's second law:
$$F_{net} = \sum F_y = T_{cord} - mg = ma_y$$For Q1, the elevator has an upward acceleration $a_y = a$ and the tension is $T$.
$$T - mg = ma$$Solving for the mass $m$:
$$T = m(g+a) \implies m = \frac{T}{g+a}$$For Q2, the elevator also has an upward acceleration $a_y = a$. The equation for the new tension $T'$ is:
$$T' - mg = ma$$ $$T' = m(g+a)$$Since the acceleration is identical in both cases, the net force must be the same. Therefore, the tension $T'$ must be equal to the original tension $T$.