Question

A 14.0 kg bucket is lowered vertically by a rope in which there is 163 N tension at a given instant. What is the acceleration of the bucket? Is it upward or downward?

Short answer

The acceleration of the bucket is $1.84\frac{\mathrm{m}}{{\mathrm{s}}^2}$, and the direction of the acceleration is upward.

Step-by-step solution

Step 1. Given data and assumption

When any object is moving upward with a constant acceleration, the rope's tension is greater than the object's weight.

Given data:

The mass of the bucket is $m=14.0\mathrm{kg}$.

The tension in the rope is $T=163\mathrm{N}$.

Assumption:

Let a be the acceleration of the bucket.

Step 2. Calculation of the acceleration of the bucket 

The weight of the bucket is

$\begin{array}{c}W=mg\\ =\left(14.0\mathrm{kg}\right)\times \left(9.80\frac{\mathrm{m}}{{\mathrm{s}}^2}\right)\\ =137.20\mathrm{N}\end{array}$

As the tension in the rope is greater than the weight, the bucket is accelerating upward.

Now, using Newton's second law,

$\begin{array}{c}ma=T-mg\\ a=\frac{163\mathrm{N}-137.20\mathrm{N}}{14.0\mathrm{kg}}\\ =1.84\frac{\mathrm{m}}{{\mathrm{s}}^2}\end{array}$

Hence, the acceleration of the bucket is $1.84\frac{\mathrm{m}}{{\mathrm{s}}^2}$, and the direction of the acceleration is upward.