# Motion - Dynamics

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Dynamics is the study of why things move, in contrast to kinematics, which is concerned with describing the motion. Things move because they are subjected to forces. A force applied to an object causes an acceleration proportional to the object's mass:F = ma. In this equation, Newton's Second Law, F and a are both vectors. The units have to be consistent.

## Newton's 1st Law of Motion

Objects at rest tend to stay at rest unless acted upon by an external unbalanced force. Objects moving with uniform motion(constant velocity in a straight line) will continue to do so unless acted upon by an external unbalanced force.

This law makes it easy to understand certain situations. For example, consider a bus moving steadily at 100 km/h along a highway. What is the force acting on it? Many forces are acting - gravity, air resistance, friction with the highway, and so on. However, we know immediately that all of them cancel out because uniform motion means NO net external force acting on the bus.

It helps understand what happens to a passenger in a car when the car has a collision and stops suddenly. The passenger continues in uniform motion until a seat belt or windshield exerts an external force to decellerate him.

A good experiment to illustrate the first law is to put a book on a skateboard. Accelerate them up to a good speed, then suddenly stop the skateboard. The book will continue on if its friction force with the board is small

## Newton's 2nd Law of Motion

Newton's 2nd Law of Motion states:

"the rate of change of the momentum of an object is directly proportional to the resultant force acting upon it".

Written mathematically, this gives

$F = \frac{dp}{dt}$

this can easily be reduced to F = ma by using differentials

$F = \frac{dp}{dt} = m\frac{dv}{dt} + v\frac{dm}{dt}$

in a constant mass system

$v\frac{dm}{dt} = 0$

and so

$F = m\frac{dv}{dt} = ma$

as

$a = \frac{dv}{dt}$

Therefore, in a system of constant mass, the acceleration of an object is directly proportional to the resultant force acting upon it.

This formula is an experimental result. You can find it for yourself if you have some means of measuring acceleration, such as a tickertape timer or sonar attachment for a calculator to measure velocity continuously.

In the experiment, various forces are applied to a wheeled cart or glider on a frictionless air track. The easiest way to get a known force is to use the force of gravity on a hanging weight with a pulley to change the force from horizontal to vertical. Each 100 grams of mass has a force of gravity of about 1 Newton.

The tickertape is pulled through a timer that marks a dot on it every tenth of a second. From the distance between dots, the velocity can be calculated. From the change in velocity from one pair of dots to the next, the acceleration can be calculated. The experiment is repeated with various hanging masses causing different pulling forces and the acceleration is measured from the recorded motion. Graphing the experimental values for the applied force versus the resulting accelerations produces a straight line graph to within experimental accuracy. The slope is equal to the mass of the moving system to within experimental error. The formula for the graph is therefore F = ma. Force has units of Newtons where a Newton (N) is equal to a kg·m/s².

### Example 1

A 1000 kg car accelerates at 0.5 m/s². Calculate the force that must be applied to it.

Solution
\begin{align} F & = ma \\ & = 1000 \mathrm{kg} \times 0.5 \text{m/s}^2 \\ & = 500 \text{kg} \cdot \text{m/s}^2 \\ & = 500 \mathrm{N} \end{align}

### example 2

A 0.085 kg bullet is fired from a rifle and emerges with a speed of 400 m/s. Assuming that the bullet has constant acceleration over the 0.5 m length of gun barrel, calculate the force on the bullet.

Solution: This is one of those tricky problems when the time is not known so we can't use a = Δv/Δt. We could sketch a v vs t graph where we know the area beneath is the distance 0.5 m:

Knowing the time for the bullet to accelerate in the rifle, we can find the acceleration and then the force:

\begin{align} F & = ma \\ & = m \frac{\Delta v}{\Delta t} \\ & = 0.085 \frac{400}{0.0025} \\ & = 13600 \mathrm{N} \end{align}

## Newton's 3rd Law of Motion

If object A applies a force on object B, then B will apply an equal force on A in the opposite direction. For example, if a car accelerates due to a 500 N force on it there must be another object that feels a 500 N force in the opposite direction - the road. An aircraft accelerates because it pushes air backward, and the air then pushes the aircraft forward. In space, where there is no air, a rocket must be used - it pushes backward on its exhaust material and the exhaust pushes forward on the rocket.

Often there are multiple forces involved. When a person shoots a rifle, a force is obviously applied to the bullet. It is the hot gunpowder gases that push on the bullet, and the bullet pushes back on the gases. The gases push back on the gun and the gun pushes forward on the gases. The gun pushes back on the person holding it, and the person pushes forward on the gun. You could continue this to the person pushing back on the Earth and the Earth pushing forward on the gun.

The Earth feels many forces from millions of cars and billions of people pushing when they accelerate. Most of these forces are canceled out when the cars or persons stop and all average out because everything is pushing different directions.

The Earth and a ball in the air exert equal and opposite forces on each other.