Conceptual Questions related to Newton’s Third Law of Motion

Que: A horse pulls forward on a carriage with a given force. By Newton's Third Law, the carriage must be pulling on the horse backward with an equal and opposite force. Given this, what explains why the horse and carriage can move forward?

The forward force of the horse is just big enough to overcome the backward force of the cart and start the cart forward
The cart's force is only in reaction to the horse's force so it does not define direction of movement
There is a brief moment where the horse pulls before the reaction force kicks in
The forward and backward forces are equal, so it actually can't move forward
The cart is rolling on wheels while the horse's hooves have traction with the ground (true)

The two forces definitely are equal and are concurrent. The moment that the ropes/attachments between the horse and cart begin to tug the cart forward, they will also pull back on the horse.
As a thought experiment, imagine if the cart had no wheels and just sat on the ground. Then the horse would have to have to apply much more force to overcome the force of friction between the cart and the ground (with wheels, the horse has to just overcome the friction between the wheels and the axles which are probably lubricated).
Now imagine if the cart were tightly attached to the ground somehow. Then the horse wouldn't be able to move the cart at all.
What is really happening if you think about the whole horse-cart system, is that the horses feet are pushing down and backwards against the ground which is pushing up and forward (the friction is really what is pushing forward as the foot attempts to go backward) against the feet. This interaction is what allows the horse to generate the forward force in the first place. The only forces resisting this forward motion are the forces of friction in the inner workings of the cart's wheel-axle system.

Que: Which best explains why we are able to accelerate forward when starting to run?
The foot not touching the ground propels the entire body as it swings forward.
The runner's upper body quickly leans forward, causing the entire body to begin accelerating forward.
The striking foot pushes backward against the ground. The friction with the ground provides an equal and opposite force forward. (true)
As one leg moves backward, it provides an equal and opposite force for the other foot to move forward.
No acceleration takes place. Runners are always at a fixed velocity.

In order for the runner to accelerate forward, a net force has to be applied to the runner. The only object making contact with the runner is the ground. As the runner's striking foot attempts to rub backward against the ground, the force of friction keeps it "in place" (relative to the ground) by providing an equal and opposite force forward.
Think about what would happen if you tried to run barefoot on ice (other than getting frostbite on your feet).
If there were no friction between the foot/shoe and the ground, it would be impossible to "run."

Que: If you are an astronaut slowly drifting away from the space station, you might be able to drift back to the station by throwing a 55-kg tool rapidly in the direction that you are traveling (away from the station).

To throw the tool, you'll apply a force on the tool directed away from the station. By Newton's third law, the tool will apply an equal and opposite force on you. If this force is large enough and applied long enough, it may be able to change the direction of your velocity towards the station.

Que: Gravity is pulling on you downwards with a force which we call your weight. The reason why you aren't accelerating downwards is that there is an equal and opposite force of the floor (let's assume you are standing up) pushing you upwards that nets out against the force of gravity. This is the "equal and opposite" force described by Newton's Third Law of Motion.

It is true that the upwards force of the floor nets against the force of gravity, but this is not the "equal and opposite" force described by Newton's Third Law.
The "equal and opposite" force described by the Third Law is the force of gravity pulling the Earth towards you (the equal and opposite forces in the third law act on the two bodies interacting, not just on one body)

Que: You and a friend are pulling on a rope in opposite directions as hard as you can. What is the "equal and opposite force" to the force of your hand pulling on the rope described by Newton's Third Law?
The force of your arm pulling back on your hand
The force of friction between the ground and your shoes
The force of the rope pulling your friend's hand
The force of the rope pulling on your hand in the opposite direction (true)
The force of your friend pulling on the rope in the opposite direction

If object A applies a force to object B, then the "equal and opposite force" is the force that B applies to A (same magnitude, but opposite direction). They don't net out with each other because they are acting on two different bodies (when we net forces, we are talking about forces on the same body).
Newton's Third Law, therefore, is describing the force of the rope pulling on your hand in the opposite direction.

Que: When you step on a roach, the roach applies an equal and opposite force on your foot as your foot applies to the roach (we don't advocate killing insects this way, especially if you are barefoot).

This comes directly from Newton's Third Law. The only reason that the roach gets squished and your foot doesn't is that your foot is less delicate.

Que: When a large truck pushes a small car with a given force, the small car is applying an equal and opposite force on the truck.

This is absolutely true and is an example of Newton's Third Law. This is why even the large truck's bumper could be compressed if it pushes with too large a force.

Que: Which of Newton's Laws gives the reason for why you can feel things that you touch?
First Law
Second Law
Third Law

The reason why you feel something that you touch (where you are applying a very small force), is explained by Newton's Third Law. The thing you are touching is applying an equal and opposite force to your hand (which compresses your finger ever so slightly which activates your "touch" sensors).

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