Dynamics
Beginner
Newton's Law
Source: Principles of Physics
Problem
An elevator cab of weight $W$ moves upward.
- What is the tension $T_{up}$ in the cable if the cab's speed is increasing at a rate of $a$?
- What is the tension $T_{down}$ in the cable if the cab's speed is decreasing at a rate of $a$?
[Q1] $T_{up} = W(1 + a/g)$ [Q2] $T_{down} = W(1 - a/g)$
Let the upward direction be positive. The forces acting on the elevator cab are the upward tension $T$ and the downward weight $W$. The mass of the cab is $m = W/g$.
Applying Newton's second law, $\sum F_y = ma_y$:
$$T - W = ma_y$$Solving for tension $T$ and substituting $m = W/g$:
$$T = W + ma_y = W + \frac{W}{g}a_y = W\left(1 + \frac{a_y}{g}\right)$$For [Q1], the cab is accelerating upward, so $a_y = +a$.
$$T_{up} = W\left(1 + \frac{a}{g}\right)$$For [Q2], the cab is moving upward but its speed is decreasing, which means its acceleration is downward. So, $a_y = -a$.
$$T_{down} = W\left(1 - \frac{a}{g}\right)$$