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# Newtons Second Law

The relationship between force, mass and acceleration of an object.
Contents

## Overview

Newton's second law of motion states that the rate of change of momentum of an object is directly proportional to the force applied, and occurs in the direction of the force; where momentum is the product of the mass of the object and its velocity. Therefore a small force is required to propel a small bottle rocket through the air, yet a proportionately much larger amount of force will be required to accelerate a full size rocket. In both cases, in the absence of an external unbalanced force, the force applied to the rocket will be proportional to its rate of change of momentum.

## Description

Newton's second law defines the relationship between
• Force
• Mass
• Acceleration
It is usually defines as:
$F=m&space;\cdot&space;a$

Example:
[metric]
##### Example - Car and Trailer
Problem
A car of mass 800kg is pulling a trailer of mass 200kg along a straight road. The car is connected to the trailer by a tow bar which can be modelled by a light inextensible rod. The driving force of the car is 2.6kN and the total resistance to motion of the car and trailer is 600N.

1. Determine the acceleration of the system.
2. Given that the reistances of the car and trailer are proportional to their masses. Determine the tension in the tow bar
3. The driver sees an accident ahead and applies the brakes causing the car to decelerate at $\inline&space;3m/s^2$. Determine the braking force applied and the force acting in the tow bar.
Workings
The total mass is $\inline&space;800&space;+&space;200&space;=&space;1000&space;kg$.

Newtons Second Law states that $\inline&space;\text{Force}=\text{Mass}&space;\cdot&space;\text{Acceleration}$

$\therefore\;\;\;\;\;26000&space;-&space;600&space;=&space;2000\[=&space;1000\times&space;\text{Acceleration}$
Thus Acceleration $\inline&space;\displaystyle&space;=&space;\frac{2000}{1000}&space;=&space;2\;m/s^2$

<h4>Tow bar tension</h4>

As the resistance to motion of the car and trailer are proportional to the masses, then

• The Resistance of the car is $\inline&space;=\frac{800}{1000}\times&space;600&space;=&space;480\;N$
• The Resistance of the traier is $\inline&space;=\frac{200}{1000}\times&space;600&space;=&space;120\;N$

By Newton's Second Law
$F=m&space;\times&space;a$
$T&space;-&space;120&space;=&space;200\times&space;2$
Therefore the tension in the towbar is $\inline&space;T=&space;400&space;+&space;120&space;=&space;520\;&space;N$

Instead of considering the Trailer we could have applied Newton's Second Law to the Car. In which case:

$2600&space;-&space;T&space;-&space;480&space;=&space;800\times&space;2&space;=&space;1600&space;Newtons$
$\therefore\;\;\;\;\;T&space;=&space;1600&space;+&space;480&space;-&space;2600&space;=&space;520\;N$

i.e. Exactly the same as before.

<h4>The car now slows down</h4>

We are told that the car now decelerates at $\inline&space;3&space;m/s^2$ i.e. its has a negative acceleration.

As we have taken the direction of right to left as positive we will stick to this but in practice provided that due care is taken with the signs, it is not important.

Consider the Trailer

The decelerating force is the force through the tow bar and the resistance to motion.

Applying Newton's Second Law:
$-&space;T&space;-&space;120\;=&space;-&space;3\times&space;200$
$\therefore\;\;\;\;\;T&space;=&space;(3\times&space;200)&space;-&space;120&space;=&space;480\;N$

We could have done this calculation by working through the forces on the car. In which case we would have needed to discover the magnitude of the decelerating force. To do this we would need to apply Newtons Second law to the combine car and trailer.

$-&space;\text{Force}&space;-&space;600&space;+T&space;=&space;-&space;3\times&space;1000\;N$
Therefore the Breaking Force $\inline&space;=&space;3000&space;-&space;600&space;=&space;2400\;N$

Now Applying Newton's Second Law to the Car alone:

$-&space;2400&space;-&space;480&space;+&space;T&space;=&space;-3\times&space;800$
Therefore
$T=&space;-&space;2400&space;+&space;2400&space;+&space;480&space;=&space;480\;N$

Solution
1. Acceleration $\inline&space;\displaystyle&space;=&space;\frac{2000}{1000}&space;=&space;2\;m/s^2$.
2. Tension in tow bar is 520N.
3. Breaking Force is 2400N. Tension in tow bar is 480N.
References