6.1 Solving problems with newton’s laws  (Page 4/12)

Particle acceleration

We have given a variety of examples of particles in equilibrium. We now turn our attention to particle acceleration problems, which are the result of a nonzero net force. Refer again to the steps given at the beginning of this section, and notice how they are applied to the following examples.

Drag force on a barge

Strategy

The drag of the water ${\overrightarrow{F}}_{\text{D}}$ is in the direction opposite to the direction of motion of the boat; this force thus works against ${\overrightarrow{F}}_{\text{app}},$ as shown in the free-body diagram in [link] (b). The system of interest here is the barge, since the forces on it are given as well as its acceleration. Because the applied forces are perpendicular, the x - and y -axes are in the same direction as ${\overrightarrow{F}}_1$ and ${\overrightarrow{F}}_2.$ The problem quickly becomes a one-dimensional problem along the direction of ${\overrightarrow{F}}_{\text{app}}$ , since friction is in the direction opposite to ${\overrightarrow{F}}_{\text{app}}.$ Our strategy is to find the magnitude and direction of the net applied force ${\overrightarrow{F}}_{\text{app}}$ and then apply Newton’s second law to solve for the drag force ${\overrightarrow{F}}_{\text{D}}.$

Solution

The angle is given by

From Newton’s first law, we know this is the same direction as the acceleration. We also know that ${\overrightarrow{F}}_{\text{D}}$ is in the opposite direction of ${\overrightarrow{F}}_{\text{app}},$ since it acts to slow down the acceleration. Therefore, the net external force is in the same direction as ${\overrightarrow{F}}_{\text{app}},$ but its magnitude is slightly less than ${\overrightarrow{F}}_{\text{app}}.$ The problem is now one-dimensional. From the free-body diagram, we can see that

However, Newton’s second law states that

Thus,

This can be solved for the magnitude of the drag force of the water $F_{\text{D}}$ in terms of known quantities:

Substituting known values gives