2.3 Newton's Third Law

Paired Forces Between Objects

Newton's Third Law is a fundamental principle that describes how forces interact between objects. When two objects interact, they apply forces to each other that are equal in magnitude and opposite in direction. This relationship forms the basis for understanding countless physical phenomena.

$\vec{F}_{\mathrm{A} \text { on } \mathrm{B}}=-\vec{F}_{\mathrm{B} \text { on } \mathrm{A}}$

The negative sign in this equation indicates that the forces point in opposite directions. These paired forces always act on different objects, which is a crucial distinction to understand.

• When you push on a wall, the wall simultaneously pushes back on you with an equal force
• When a hammer strikes a nail, the nail exerts an equal force back on the hammer
• As a swimmer pushes water backward, the water pushes the swimmer forward

These paired forces do not cancel each other out because they act on different objects. This explains why you can't lift yourself up by pulling on your shoelaces - both forces act on you, creating a net force of zero.

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Representing Third-Law Pairs

To represent a Newton's third-law pair, draw the two interacting objects separately and label the forces as $\vec{F}_{A\text{ on }B}$ on object B and $\vec{F}_{B\text{ on }A}$ on object A. These two forces are equal in magnitude and opposite in direction, but they appear on different free-body diagrams because they act on different objects.

For example, if a person pushes a wall, you would draw two separate diagrams: one of the wall showing $\vec{F}_{\text{person on wall}}$ pointing into the wall, and one of the person showing $\vec{F}_{\text{wall on person}}$ pointing back into the person. Each force in the pair acts on a different object, so they never appear together on the same free-body diagram. Practicing this labeling convention — always identifying "who pushes whom" — is essential for correctly applying Newton's Third Law.

Internal Forces and Center of Mass

The motion of a system can be analyzed by distinguishing between internal and external forces. This distinction helps us understand why some forces affect the overall motion while others don't.

Internal forces occur between objects within a defined system. These forces:

• Are interactions between parts of the system (like molecules in a solid)
• Always come in equal and opposite pairs according to Newton's Third Law
• Do not affect the motion of the system's center of mass

External forces involve interactions between system objects and the outside environment:

• Act on system objects from sources outside the defined system
• Can change the motion of the center of mass
• Determine the acceleration of the center of mass according to Newton's Second Law

The center of mass of a system moves as if all the system's mass were concentrated at that point and all external forces were applied there. This is why internal forces, which always come in equal and opposite pairs, cannot change the motion of the center of mass.

Tension in Strings and Cables

Tension represents the pulling force transmitted through a string, cable, or similar object when it is pulled tight by forces acting at each end. This force is essential in many mechanical systems.

Tension is the macroscopic result of microscopic electromagnetic forces between atoms that hold the string together. 🪢 When a string is pulled, these atomic bonds resist being stretched, creating the tension force we observe.

Tension can also be understood at the system level as the macroscopic net result of adjacent infinitesimal segments of a string, cable, or chain pulling on each other in response to external forces applied to the system. In other words, each tiny segment exerts a force on the neighboring segment, and this distributed interaction is what we call tension.

• At the molecular level, atoms are pulled slightly apart but resist separation
• This resistance creates a pulling force that transmits through the entire string
• The tension force always points away from a point on the string, pulling inward at both ends

In an ideal string with negligible mass, the tension is uniform throughout the entire length. This simplification allows for easier analysis of many mechanical systems.

Properties of Ideal Strings

Ideal strings are theoretical constructs that simplify problem-solving in physics. While real strings have mass and can stretch, the ideal string model provides a useful approximation for many situations.

An ideal string has these key characteristics:

• Negligible mass compared to the objects it connects
• Inelastic (does not stretch) when force is applied
• Can be bent around objects like pulleys with no friction
• Transmits tension undiminished from one end to the other

In an ideal string, the tension force has the same magnitude at all points along the string. This property allows us to analyze complex systems like pulleys with greater ease.

Tension Variations in Strings

Real strings with non-negligible mass exhibit tension variations along their length. This occurs because each segment of the string must support not only the attached load but also the weight of the string below it.

The tension in a hanging string:

• Is maximum at the top where it supports both the load and the entire string weight
• Decreases gradually toward the bottom as less string weight needs to be supported
• Reaches minimum at the bottom where it only supports the attached load

The amount of tension variation depends on the string's mass and length. For short, lightweight strings, assuming constant tension throughout is often a reasonable approximation that simplifies calculations without introducing significant error.

Ideal Pulleys

An ideal pulley is a simple machine that changes the direction of a tension force without changing its magnitude. 🧵 This allows forces to be applied in more convenient directions while maintaining their effectiveness.

Properties of an ideal pulley include:

• Negligible mass (so its rotational inertia can be neglected)
• Rotates about an axle through its center of mass
• Negligible friction at the axle

When combined with an ideal string, this model gives the same tension throughout the string while allowing the pulley to change the direction of the force.

Ideal pulleys maintain equal tension on both sides, merely redirecting the force without changing its magnitude. This property makes pulleys extremely useful in mechanical systems for lifting loads, applying forces in different directions, and transmitting power efficiently.

Practice Problem 1: Newton's Third Law Paired Forces

Solution

First, we need to apply Newton's Third Law. When the person pushes forward with 300 N to jump, the boat experiences an equal and opposite force of 300 N backward.

Using Newton's Second Law for the boat: $F = ma$

Rearranging to solve for acceleration: $a = \frac{F}{m} = \frac{300 \text{ N}}{75 \text{ kg}} = 4 \text{ m/s}^2$

Therefore, the boat initially accelerates backward at 4 m/s² when the person jumps.

Practice Problem 2: Tension in an Ideal String

Solution

Because the 5 kg mass is heavier, it accelerates downward and the 3 kg mass accelerates upward. Let downward be positive for the 5 kg mass and upward be positive for the 3 kg mass.

For the 5 kg mass: $5g - T = 5a$

For the 3 kg mass: $T - 3g = 3a$

Add the two equations: $5g - T + T - 3g = 5a + 3a$

$2g = 8a$ $a = \frac{2g}{8} = \frac{2 \times 9.8}{8} = 2.45\ \text{m/s}^2$

Now solve for the tension using either mass. Using the 3 kg mass: $T - 3g = 3a$

$T = 3g + 3a = 3(9.8) + 3(2.45) = 29.4 + 7.35 = 36.75\ \text{N}$

Therefore, the tension in the string is $36.75\ \text{N}$, and the 5 kg mass accelerates downward at $2.45\ \text{m/s}^2$ while the 3 kg mass accelerates upward at $2.45\ \text{m/s}^2$.