Problem — Crate Hanging from the Ceiling

Problem

A crate of $8{,}0\;\text{kg}$ hangs from a ceiling by a rope. What is the tension in the rope? What changes if the crate is pulled upward with acceleration $2{,}0\;\text{m/s}^2$?

Solution

Case 1 — Crate in equilibrium. The crate is at rest (or moving at constant velocity): the acceleration is zero. The tension $T$ acting upward and the weight $mg$ acting downward act on the crate and balance each other:

$T - mg = 0 \quad\Longrightarrow\quad T = mg = 8{,}0\;\text{kg} \times 9{,}8\;\text{m/s}^2 = 78{,}4\;\text{N}$

$\ev{T = 78{,}4\;\mathrm{N}}$

Case 2 — Crate pulled upward with $a = 2{,}0\;\text{m/s}^2$. Now the crate accelerates upward: the tension must not only support the weight, but also supply the net force that produces the acceleration. With the axis pointing upward,

$T - mg = m\,a \quad\Longrightarrow\quad T = m(g + a)$

$T = 8{,}0\;\text{kg} \times (9{,}8 + 2{,}0)\;\text{m/s}^2 = 8{,}0 \times 11{,}8 = 94{,}4\;\text{N}$

$\ev{T = 94{,}4\;\mathrm{N}}$

The tension increases relative to the static case: accelerating upward, the rope is more “loaded”, because it must overcome the weight and also impart the acceleration.

Links

Topics: Dynamics Concepts: Tension · Newton’s second law Objects: Ideal rope