Force and acceleration

A puzzle...

  • Objects with different masses feel a different gravitational force (weight).
  • Acceleration is caused by forces.
  • But all objects (independent of mass) fall with the *same* acceleration??
  • How can it be that objects being pushed by gravity with different forces all have the same acceleration???

What's your experience?

...with acceleration and force.

You push on a bowling ball. It accelerates.

You push *harder* on a bowling ball, how does its acceleration change?

So, writing things as an equation (where $c$ is just some constant, because we know that $a$ and $F$ have different units):

  1. $a=\frac cF$?
  2. $a=c\cdot F$?
  3. $a=c$? (Always the same acceleration, no matter what the force.)

But maybe acceleration depends not only on the force, but also on the mass? (units: kilograms)

You pull on something heavy, it accelerates.

Now you pull on something lighter with the same force. Does it accelerate more or less?

Trying to write an equation (where $k$ is just some constant, and here $F$ is constant too, because we're comparing with the same $F$.)

Which equation captures that behavior?

  1. $a=\frac{k\cdot F}{m}$?
  2. $a=k\cdot F\cdot m$?
  3. $a=k\cdot F$? (mass of the object doesn't matter)

Newton's law of acceleration

$$a=\frac Fm.$$ The units of force are "Newtons" (N) when $a$ has units of m/s${}^2$ and $m$ has units of kg.

To make the units come out right, 1 N is the same as...

Weight?

Weight (N) is the force of gravity on an object. Call it $F_g$.

It depends on the *mass*(kg) of an object. $F_g = c\cdot m$.

Now, think about Newtons law, $$a = \frac Fm$$ If the force is due to gravity, we call it the "weight" ($F_g$).

Then $F=F_g=c\cdot m$: $$a=\frac{(c\cdot m)}{m}=c$$

The acceleration of any object (neglecting air resistance) near Earth's surface is 9.8 m/s${}^2\equiv g$. So, apparently the weight of an object (in Newtons) is just:

$$\text{Weight}=F_g= mg,$$ where $g$=9.8 m/s${}^2$

Bonus: This tells us something about the weight of an object on another planet to. "All" we have to do is measure the acceleration of one falling object.

Force has a direction

Acceleration has a direction: it's the direction of the change in velocity:

speedup

Acceleration is the result of a force acting on an object, so force has the same direction as the change of velocity.

Multiple forces in action

Tug of war
There are times that we know *for sure* that a force is acting on an object, and yet there's no acceleration. ??

Force (like velocity, $\myv v$) can be thought of as having a direction. We can picture $F$ as an arrow, where the length of the arrow is the size of the force.

"Adding up the forces" amounts to lining up all the arrows head to tail...

And it's the sum of all these forces that gives rise to the acceleration. So we should write:

$$a=\frac{F_{\total}}{m}.$$

One body exerts a force on another body if, in the absence of other forces, the first body *would* cause the second body to accelerate.

Summary

The force of gravity (near Earth's surface) is: $$F_g=m g=m\cdot9.8\text{ m/s/s}$$ Therefore force has units of "kg m / s${}^2$". This unwieldy combination is given the name "Newtons": 1 N = 1 kg m / s${}^2$.

Newton's Law says that there is a relationship between any force (not just gravity), mass, and acceleration: $$a = F/m$$ When $F$ is in Newtons, and $m$ is in kilograms, then the acceleration will have units of m/s${}^2$.

Tug of war The force in the equation above is not just any one force acting on an object, but rather the sum of all the forces acting on the object.

Image credits

Christian Bachellier, Laura Dahl, Zunami