# 4.7 Further applications of newton's laws of motion

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## Learning objectives

By the end of this section, you will be able to:

• Apply problem-solving techniques to solve for quantities in more complex systems of forces.
• Integrate concepts from kinematics to solve problems using Newton's laws of motion.

The information presented in this section supports the following AP® learning objectives and science practices:

• 3.A.2.1 The student is able to represent forces in diagrams or mathematically using appropriately labeled vectors with magnitude, direction, and units during the analysis of a situation. (S.P. 1.1)
• 3.A.3.1 The student is able to analyze a scenario and make claims (develop arguments, justify assertions) about the forces exerted on an object by other objects for different types of forces or components of forces. (S.P. 6.4, 7.2)
• 3.A.3.3 The student is able to describe a force as an interaction between two objects and identify both objects for any force. (S.P. 1.4)
• 3.B.1.1 The student is able to predict the motion of an object subject to forces exerted by several objects using an application of Newton's second law in a variety of physical situations with acceleration in one dimension. (S.P. 6.4, 7.2)
• 3.B.1.3 The student is able to re-express a free-body diagram representation into a mathematical representation and solve the mathematical representation for the acceleration of the object. (S.P. 1.5, 2.2)
• 3.B.2.1 The student is able to create and use free-body diagrams to analyze physical situations to solve problems with motion qualitatively and quantitatively. (S.P. 1.1, 1.4, 2.2)

There are many interesting applications of Newton’s laws of motion, a few more of which are presented in this section. These serve also to illustrate some further subtleties of physics and to help build problem-solving skills.

## Drag force on a barge

Suppose two tugboats push on a barge at different angles, as shown in [link] . The first tugboat exerts a force of $2.7×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{N}$ in the x -direction, and the second tugboat exerts a force of $3.6×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{N}$ in the y -direction.

If the mass of the barge is $5.0×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{kg}$ and its acceleration is observed to be $7\text{.}\text{5}×{\text{10}}^{-2}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}$ in the direction shown, what is the drag force of the water on the barge resisting the motion? (Note: drag force is a frictional force exerted by fluids, such as air or water. The drag force opposes the motion of the object.)

Strategy

The directions and magnitudes of acceleration and the applied forces are given in [link] (a) . We will define the total force of the tugboats on the barge as ${\mathbf{\text{F}}}_{\text{app}}$ so that:

${\mathbf{\text{F}}}_{\text{app}}\text{=}{\mathbf{\text{F}}}_{\mathit{x}}+{\mathbf{\text{F}}}_{\mathit{y}}$

Propose a force standard different from the example of a stretched spring discussed in the text. Your standard must be capable of producing the same force repeatedly.
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