Newton's Second Law of Motion

According to Newton's first law, things 'like to keep on doing what they are doing'. In other words, if an object is moving, it tends to continue moving (in a straight line and at the same speed) and if an object is stationary, it tends to remain stationary. So how do objects start moving?

Let us look at the example of a $\text{10}$ $\text{kg}$ box on a rough table. If we push lightly on the box as indicated in the diagram, the box won't move. Let's say we applied a force of $\text{100}$ $\text{N}$, yet the box remains stationary. At this point a frictional force of $\text{100}$ $\text{N}$ is acting on the box, preventing the box from moving. If we increase the force, let's say to $\text{150}$ $\text{N}$ and the box almost starts to move, the frictional force is $\text{150}$ $\text{N}$. To be able to move the box, we need to push hard enough to overcome the friction and then move the box. If we therefore apply a force of $\text{200}$ $\text{N}$ remembering that a frictional forceof $\text{150}$ $\text{N}$ is present, the 'first' $\text{150}$ $\text{N}$ will be used to overcome or 'cancel' the friction and the other $\text{50}$ $\text{N}$ will be used to move (accelerate) the block. In order to accelerate an object we must have a resultant force acting on the block.

Now, what do you think will happen if we pushed harder, lets say $\text{300}$ $\text{N}$? Or, what do you think will happen if the mass of the block was more, say $\text{20}$ $\text{kg}$, or what if it was less? Let us investigate how the motion of an object is affected by mass and force.

A recommended experiment for formal assessment on Newton's second law of motion is also included in this chapter. In this experiment learners will investigate the relationship between force and acceleration (Newton's second law). You will need trolleys, different masses, inclined plane, rubber bands, meter ruler, ticker tape apparatus, ticker timer, graph paper.

Experiment: Newton's Second Law of Motion

Aim

To investigate the relation between the acceleration of objects and the application of a constant resultant force.

Method

A constant force of $\text{20}$ $\text{N}$, acting at an angle of $\text{60}$$\text{°}$ to the horizontal, is applied to a dynamics trolley.
Ticker tape attached to the trolley runs through a ticker timer of frequency $\text{20}$ $\text{Hz}$ as the trolley is moving on the frictionless surface.
The above procedure is repeated 4 times, each time using the same force, but varying the mass of the trolley as follows:
1. Case 1: $\text{6.25}$ $\text{kg}$
2. Case 2: $\text{3.57}$ $\text{kg}$
3. Case 3: $\text{2.27}$ $\text{kg}$
4. Case 4: $\text{1.67}$ $\text{kg}$
Shown below are sections of the four ticker tapes obtained. The tapes are marked with the letters A, B, C, D, etc. A is the first dot, B is the second dot and so on. The distance between each dot is also shown.

Instructions

Use each tape to calculate the instantaneous velocity (in $\text{m·s$^{-1}$}$) of the trolley at points B and F (remember to convert the distances to m first!). Use these velocities to calculate the trolley's acceleration in each case.
Tabulate the mass and corresponding acceleration values as calculated in each case. Ensure that each column and row in your table is appropriately labelled.
Draw a graph of acceleration vs. mass, using a scale of $\text{1}$ $\text{cm}$ = $\text{1}$ $\text{m·s$^{-2}$}$ on the y-axis and $\text{1}$ $\text{cm}$ = $\text{1}$ $\text{kg}$ on the x-axis.
Use your graph to read off the acceleration of the trolley if its mass is $\text{5}$ $\text{kg}$.
Write down a conclusion for the experiment.

You will have noted in the investigation above that the heavier the trolley is, the slower it moved when the force was constant. The acceleration is inversely proportional to the mass. In mathematical terms:

$a\propto \frac{1}{m}$

In a similar investigation where the mass is kept constant, but the applied force is varied, you will find that the bigger the force is, the faster the object will move. The acceleration of the trolley is therefore directly proportional to the resultant force. In mathematical terms:

$a\propto F.$

Rearranging the above equations, we get a $\propto$$\frac{F}{m}$ or $F = ma$.

Remember that both force and acceleration are vectors quantities. The acceleration is in the same direction as the force that is being applied. If multiple forces are acting simultaneously then we only need to work with the resultant force or net force.

Definition: Newton's Second Law of Motion

If a resultant force acts on a body, it will cause the body to accelerate in the direction of the resultant force. The acceleration of the body will be directly proportional to the resultant force and inversely proportional to the mass of the body. The mathematical representation is:\[\vec{F}_{net} = m\vec{a}\]

Force is a vector quantity. Newton's second law of motion should be applied to the $y$- and $x$-directions separately. You can use the resulting $y$- and $x$-direction resultants to calculate the overall resultant as we saw in the previous chapter.

Newton's Second Law of Motion