63 Accelerating the Body

Newton’s Second Law of Motion

tells us that we need a in order to create an .   As you might expect, a larger net force will cause a larger acceleration, and the same net force will give a smaller mass a greater acceleration. summarizes all of that into a single equation relating the , , and :

(1)   \begin{equation*} \bold{F_{net}} = m\bold{a} \end{equation*}

Finding Acceleration from Net Force

If we know the net force and want to find the acceleration, we can solve for the acceleration:

(2)   \begin{equation*} \bold{a} = \frac{\bold{F_{net}}}{m} \end{equation*}

Now we see that larger create larger and larger masses reduce the size of the acceleration. In fact, an object’s is a direct measure of an objects resistance to changing its motion, or its .

Reinforcement Exercises

Finding Net Force from Acceleration

Everyday Example: Parachute Opening

In the previous chapter we found that if opening a parachute slows a skydiver from 54 m/s to 2.7 m/s in just 2.0 s of time then they experienced an average upward acceleration of 26 m/s/s . If the mass of our example skydiver is 85 kg, what is the  average net force on the person?

We start with Newton’s Second Law of Motion

    \begin{equation*} \bold{F_{net}} = m\bold{a} \end{equation*}

Enter in our values:

    \begin{equation*} \bold{F_{net}} = (85\,\bold{kg})(26\,\bold{m/s/s}) = 2200 \,\bold{N} \end{equation*}

The person experiences an average net force of 2200 N upward during chute opening. When the chute begins to open the body position changes to feet first, which significantly reduces air resistance, so air resistance is no longer balancing the body weight. Therefore, the harness needs to support body weight plus provide the additional unbalanced 2200 N upward force on the person. The skydiver’s body weight is Fg = 85 kg x 9.8 m/s/s = 833 N, so the force on them from the harness must be 2833 N. That force is actually more than three times greater than their body weight, but is distributed over the wide straps that make up the leg loops and waist loop of the harness, which helps to prevent injury.

Reinforcement Exercises

Check out this simulation to see how forces combine to create net forces and accelerations:

Forces and Motion: Basics

Free-Fall Acceleration

In the absence of air resistance, heavy objects do not fall faster than lighter ones and all objects will fall with the same acceleration. Need experimental evidence? Check out this video:

It’s an interesting quirk of our universe that the same property of an object, specifically its , determines both the on it and its resistance to accelerations, or . Said another way,  the inertial mass and the gravitational mass are equivalent. That is why we the free-fall acceleration for all objects has a of 9.8 m/s/s, as we will show in the following example.

Everyday Example: Free-Falling

Let’s calculate the initial of our example skydiver the moment they jump. At this moment they have the pulling them down, but they have not yet gained any , so the () is zero. The is then just gravity, because it is the only force, so they are in free-fall for this moment. Starting with :

(3)   \begin{equation*} \bold{a} = \frac{\bold{F_{net}}}{m} \end{equation*}

Gravity is the net force in this case because it is the only force, so we just use the formula for calculating force of gravity near the surface of Earth, add a negative sign because down is our negative direction (\bold{F_g} = -mg), and enter that for the net force: :

(4)   \begin{equation*} \bold{a} = \frac{-mg}{m} \end{equation*}

We see that the cancels out,

(5)   \begin{equation*} \bold{a} = \frac{-\cancel{m}g}{\cancel{m}} = -g = -9.8 \bold{\frac{m}{s^2}} \end{equation*}

We see that our is negative, which makes sense because the acceleration is downward. We also see that the size, or magnitude, of the acceleration is g = 9.8 m/s2. We have just shown that in the absence of , all objects falling near the surface of Earth will experience an  equal in size to  9.8 m/s2, regardless of their and . Whether the free-fall acceleration is -9.8 m/s/s  or +9.8 m/s/s depends on if you chose downward to be the negative or positive direction.

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