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The relationship between force, mass and acceleration of an object

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Newton's second law defines the relationship between

Force
Mass
Acceleration

It is usually defines as:

$F=m \cdot a$

(1)

http://www.amazon.com/Newtons-second-motion-Reverend-Ludlam/dp/1140735780/ref=sr_1_3?s=books&ie=UTF8&qid=1316274711&sr=1-3 " An essay on Sir Isaac Newton's second law of motion. By the Reverend Mr. Ludlam."

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http://www.amazon.com/Introductory-Physics-Algebra-Second-Language/dp/0471762504/ref=sr_1_2?s=books&ie=UTF8&qid=1316283551&sr=1-2 " Introductory Physics with Algebra as a Second Language: Mastering Problem-Solving by Stuart E. Loucks."

http://www.amazon.com/Theoretical-Concepts-Physics-Alternative-Reasoning/dp/052152878X/ref=sr_1_9?s=books&ie=UTF8&qid=1316283551&sr=1-9 " Theoretical Concepts in Physics: An Alternative View of Theoretical Reasoning in Physics by M. S. Longair (Dec 22, 2003) "

Example 1 [metric]

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.

Determine the acceleration of the system.
Given that the reistances of the car and trailer are proportional to their masses. Determine the tension in the tow bar
The driver sees an accident ahead and applies the brakes causing the car to decelerate at $3m/s^2$ . Determine the braking force applied and the force acting in the tow bar.

Workings

The total mass is $800 + 200 = 1000 kg$ .

Newtons Second Law states that $\text{Force}=\text{Mass} \cdot \text{Acceleration}$

$\therefore\;\;\;\;\;26000 - 600 = 2000\[= 1000\times \text{Acceleration}$

(2)

Thus Acceleration $\displaystyle = \frac{2000}{1000} = 2\;m/s^2$

<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 $=\frac{800}{1000}\times 600 = 480\;N$
The Resistance of the traier is $=\frac{200}{1000}\times 600 = 120\;N$

23287/Newton%27s%20Second%20Law%20t2.png

By Newton's Second Law $F=m \times a$

$T - 120 = 200\times 2$

(3)

Therefore the tension in the towbar is $T= 400 + 120 = 520\; N$

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

$2600 - T - 480 = 800\times 2 = 1600 Newtons$

(4)

$\therefore\;\;\;\;\;T = 1600 + 480 - 2600 = 520\;N$

(5)

i.e. Exactly the same as before.

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

We are told that the car now decelerates at $3 m/s^2$ 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:

$- T - 120\;= - 3\times 200$

(6)

$\therefore\;\;\;\;\;T = (3\times 200) - 120 = 480\;N$

(7)

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.