Man Pulling a Box Example

A man is pulling a \(\text{20}\) \(\text{kg}\) box with a rope that makes an angle of \(\text{60}\)\(\text{°}\) with the horizontal. If he applies a force of magnitude \(\text{150}\) \(\text{N}\) and a frictional force of magnitude \(\text{15}\) \(\text{N}\) is present, calculate the acceleration of the box.

Step 1: Draw a force diagram or free body diagram

The motion is horizontal and therefore we will only consider the forces in a horizontal direction. Remember that vertical forces do not influence horizontal motion and vice versa.

Step 2: Calculate the horizontal component of the applied force

We first need to choose a direction to be the positive direction in this problem. We choose the positive \(x\)-direction (to the right) to be positive.

The applied force is acting at an angle of \(\text{60}\)\(\text{°}\) to the horizontal. We can only consider forces that are parallel to the motion. The horizontal component of the applied force needs to be calculated before we can continue: \begin{align*} F_x & = F_{applied}\cos(\theta) \\ &= \text{150}\cos(\text{60}\text{°}) \\ & = \text{75}\text{ N} \end{align*}

Step 3: Calculate the acceleration

To find the acceleration we apply Newton's second law: \begin{align*} F_R&=ma \\ F_x+F_f&=(\text{20})a \\ (\text{75}) + (-\text{15})&= (\text{20})a\\ a & = \frac{\text{60}}{\text{20}} \\ a & = \text{3}\text{ m·s$^{-2}$} \end{align*}The acceleration is \(\text{3}\) \(\text{m·s$^{-2}$}\) to the right.