# 3.8 Newton’s third law of motion: symmetry in forces  (Page 3/5)

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Solution

Newton’s second law is given by

$a=\frac{{F}_{\text{net}}}{m}.$

The net external force on System 1 is deduced from [link] and the discussion above to be

${F}_{\text{net}}={F}_{\text{floor}}-f=\text{150 N}-\text{24}\text{.}\text{0 N}=\text{126 N}.$

The mass of System 1 is

$m=\left(\text{65}\text{.}\text{0}+\text{12}\text{.}\text{0}+\text{7}\text{.}0\right)\phantom{\rule{0.25em}{0ex}}\text{kg}=\text{84 kg}.$

These values of ${F}_{\text{net}}$ and $m$ produce an acceleration of

$\begin{array}{}a=\frac{{F}_{\text{net}}}{m},\\ a=\frac{1\text{26 N}}{\text{84 kg}}=\text{1}\text{.}{\text{5 m/s}}^{2}\text{.}\end{array}$

Discussion

None of the forces between components of System 1, such as between the professor’s hands and the cart, contribute to the net external force because they are internal to System 1. Another way to look at this is to note that forces between components of a system cancel because they are equal in magnitude and opposite in direction. For example, the force exerted by the professor on the cart results in an equal and opposite force back on her. In this case both forces act on the same system and, therefore, cancel. Thus internal forces (between components of a system) cancel. Choosing System 1 was crucial to solving this problem.

## Force on the cart—choosing a new system

Calculate the force the professor exerts on the cart in [link] using data from the previous example if needed.

Strategy

If we now define the system of interest to be the cart plus equipment (System 2 in [link] ), then the net external force on System 2 is the force the professor exerts on the cart minus friction. The force she exerts on the cart, ${\mathbf{\text{F}}}_{\text{prof}}$ , is an external force acting on System 2. ${\mathbf{\text{F}}}_{\text{prof}}$ was internal to System 1, but it is external to System 2 and will enter Newton’s second law for System 2.

Solution

Newton’s second law can be used to find ${\mathbf{\text{F}}}_{\text{prof}}$ . Starting with

$a=\frac{{F}_{\text{net}}}{m}$

and noting that the magnitude of the net external force on System 2 is

${F}_{\text{net}}={F}_{\text{prof}}-f,$

we solve for ${F}_{\text{prof}}$ , the desired quantity:

${F}_{\text{prof}}={F}_{\text{net}}+f.$

The value of $f$ is given, so we must calculate net ${F}_{\text{net}}$ . That can be done since both the acceleration and mass of System 2 are known. Using Newton’s second law we see that

${F}_{\text{net}}=\text{ma},$

where the mass of System 2 is 19.0 kg ( $m$ = 12.0 kg + 7.0 kg) and its acceleration was found to be $a=1.5\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}$ in the previous example. Thus,

${F}_{\text{net}}=\text{ma},$

${F}_{\text{net}}=\left(\text{19}\text{.}\text{0 kg}\right)\left(1.5\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}\right)=\text{29 N}.$

Now we can find the desired force:

${F}_{\text{prof}}={F}_{\text{net}}+f,$

${F}_{\text{prof}}=\text{29 N}+\text{24.0 N}=\text{53 N}.$

Discussion

It is interesting that this force is significantly less than the 150-N force the professor exerted backward on the floor. Not all of that 150-N force is transmitted to the cart; some of it accelerates the professor.

The choice of a system is an important analytical step both in solving problems and in thoroughly understanding the physics of the situation (which is not necessarily the same thing).

## Phet explorations: gravity force lab

Visualize the gravitational force that two objects exert on each other. Change properties of the objects in order to see how it changes the gravity force.

## Section summary

• Newton’s third law of motion    represents a basic symmetry in nature. It states: Whenever one body exerts a force on a second body, the first body experiences a force that is equal in magnitude and opposite in direction to the force that the first body exerts.
• A thrust    is a reaction force that pushes a body forward in response to a backward force. Rockets, airplanes, and cars are pushed forward by a thrust reaction force.

## Conceptual questions

When you take off in a jet aircraft, there is a sensation of being pushed back into the seat. Explain why you move backward in the seat—is there really a force backward on you? (The same reasoning explains whiplash injuries, in which the head is apparently thrown backward.)

A device used since the 1940s to measure the kick or recoil of the body due to heart beats is the “ballistocardiograph.” What physics principle(s) are involved here to measure the force of cardiac contraction? How might we construct such a device?

Describe a situation in which one system exerts a force on another and, as a consequence, experiences a force that is equal in magnitude and opposite in direction. Which of Newton’s laws of motion apply?

Why does an ordinary rifle recoil (kick backward) when fired? The barrel of a recoilless rifle is open at both ends. Describe how Newton’s third law applies when one is fired. Can you safely stand close behind one when it is fired?

An American football lineman reasons that it is senseless to try to out-push the opposing player, since no matter how hard he pushes he will experience an equal and opposite force from the other player. Use Newton’s laws and draw a free-body diagram of an appropriate system to explain how he can still out-push the opposition if he is strong enough.

Newton’s third law of motion tells us that forces always occur in pairs of equal and opposite magnitude. Explain how the choice of the “system of interest” affects whether one such pair of forces cancels.

## Problem exercises

What net external force is exerted on a 1100-kg artillery shell fired from a battleship if the shell is accelerated at $2\text{.}\text{40}×{\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}$ ? What is the magnitude of the force exerted on the ship by the artillery shell?

Force on shell: $2\text{.}\text{64}×{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}\text{N}$

Force exerted on ship = $-2\text{.}\text{64}×{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}\text{N}$ , by Newton’s third law

A brave but inadequate rugby player is being pushed backward by an opposing player who is exerting a force of 800 N on him. The mass of the losing player plus equipment is 90.0 kg, and he is accelerating at $1\text{.}\text{20}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}$ backward. (a) What is the force of friction between the losing player’s feet and the grass? (b) What force does the winning player exert on the ground to move forward if his mass plus equipment is 110 kg? (c) Draw a sketch of the situation showing the system of interest used to solve each part. For this situation, draw a free-body diagram and write the net force equation.

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