Forces

A force is a push, a pull or a twist. In most cases, an object will have more than one force acting on it, These forces add up to produce a  total force called the net force.

In Year 10 you only have to work out the net force  for forces acting in a straight line with each other. In these cases, two forces acting in the same direction add up to produce a net force that is the total of the two. 

If the two forces are in the opposite direction then the net force is in the direction of the larger force, and the size of the force is the larger force take away the smaller one. This should not be difficult, as it makes intuitive sense from many everyday experiences. The example below illustrates this:

They need to fall very fast for air resistance to produce this much friction on a relatively small area. To slow their fall, the parachute is opened.  

When the parachute is open, the friction force will INITIALLY be larger than the gravitational force, so they will slow. As they slow, the friction force decreases. When the two forces are equal, they will once again be falling at a steady speed - but this time a much slower one.

Although a net force of zero means an object cannot be accelerating (it must be at rest or moving in a straight line at a steady speed), applying a force to an object will cause it to change shape regardless of whether the forces are balanced or not. In many cases, such as standing on a concrete slab floor, the deformation is too small for our senses to detect (such deformation can be undesirable, which is why some very high performance bearings are made of diamond). 

Force-mass-acceleration relationship

You should know from previous learning that mass is a measure of the amount of substance in an object. Its units are kilograms , kg. An objects mass in kg is NOT its weight (we will get to that later).

For a given mass, the amount of acceleration it experiences is directly related to the force: twice the force, twice the acceleration, and so on. The unit of force, the newton, has therefore been defined as the amount of net force that would cause an object of 1 kg to accelerate at 1 m s-2 .

The symbol for newton is N, so we could write 1 N = 1 kg m s-2 . 

This relationship can be written in a mathematical formula as

Fnet = m a

i.e. net force (in newtons) = mass (in kilograms) times acceleration (in metres per second squared)

Like our other formulae, it can be rearranged using the triangle or other suitable methods:

• a = Fnet/m                          m = Fnet/a 

Example: A 1200 kg car travelling at 24 m s-1 brakes suddenly and comes to a halt in 3 seconds. Calculate the unbalanced force acting on the car.

Solution: step 1 is to calculate the acceleration: a = Δv/Δt  

 Δv = 0 m s-1   to 24 m s-1 = 24 m s-1 

a = Δv/Δt= 24 m s-1/ 3 s = - 8 m s-2 

Step 2 is to use the calculated acceleration to find the force:

• F = m a  =  1200 kg x  8 m s-2      =  9600 N 

Gravity

Gravity exerts a force of approximately 10 newtons on each kilogram of mass at the Earth's surface (the number is closer to 9.8, but 10 is the value you will be given on the NCEA exam).

This will be given to you as a value on the formula sheet as:   g = 10 N kg-1

The name for the force caused by gravity is weight. Hence, if you are given the mass of an object and are asked its weight, you need to apply the value of g and give your answer in newtons e.g.

Motion

An object is either at rest or moving. If it is moving, it has an instantaneous speed. The speed shown on the speedo of your car is instantaneous speed. If the reading on the speedo stays the same,  the car is travelling at a constant speed.  NCEA level one formula problems involve objects travelling at a constant speed, or at constant acceleration,  in a straight line in one direction only.  Level 1 problems do not involve objects that turn around and go backwards, or objects that go in curves. 

An object on a journey may stop or speed up or slow down. However, by using the total distance travelled and the time it takes we can work out the average speed with the formula:

• average speed = total distance traveled/time taken

Reminder: a slash in text means "divided by" and is also the same as a fraction line in a maths formula

If the movement is at a constant (steady) speed, average speed is the same as instantaneous speed. If the object speeds up, slows down or stops it isn't.

This relationship can be written as a formula shown on the left. The three ways of rearranging this are shown below. The standard units of speed for most problems are metres per second, written in negative index notation as

metres per second   = m s-1  

note the space between the m and the s; this means "times" so this is "metres times seconds to the power of negaitive one". Making a power negative is the same as a positive power on the bottom line of a fraction. Metres per second can also be written as m/s but this way of writing can cause problems with more complex units.

Important note: NEVER write seconds as sec or secs (this will be taken as a wrong unit in your exams).

The triangle symbol is the Greek letter "delta' (the uppercase version) and means 'change in'. 

Change in distance is given by final distance - starting distance; 

delta t is calculated by final time - starting time

An object will travel in a straight line at a constant speed unless some UNBALANCED force acts on it. this is Newtn's first law of motion.  Note that something ‘not moving’ has a speed of zero. 

Acceleration

Speed can change. For example, an object could speed up from 5 m s-1 to 25 m s-1 in 5 seconds.

This means it got faster by 20 m s-1 during that 5 seconds. Its acceleration is given by 

20 m s-1 ÷ 5 s =  4 m s-2

i.e. it got faster by 4 m s-1 each second: 4 metres per second per second. We can say this as “4 metres per second squared”, and we can write it as 4 m s-2.

The formula for acceleration can be written as

this can be expressed in words as "acceleration = change in speed divided by time taken"

You can use the same triangle method to rearrange this formula (there are better ways to rearrange formulae,, though)

Graphs of motion

Motion can be graphed as distance-time graphs or speed-time graphs.(acceleration-time graphs only occur at higher levels of NCEA).

• graphs you are given will only include constant accelerations over a given time interval; this means a speed -time graph will never have curves on it

• distances for NCEA level 1 are only for an object travelling in one direction, so distance-time graphs will only ever slope upwards and never downwards

• in theory, distance-time graphs cannot have sharp corners (as this would mean instant speed change); however, you may be presented with one of these if it is drawn by someone with poor physics or drawing skills.

As a general rule, speed time graphs are easier to draw than distance time graphs. The examples below will illustrate this.  Let's look at the simple speed-time graph below: For ease of description I have divided it into three sections A, B and C because in each of those sections there is a different sort of motion.

You could interpret this in words: the skater accelerated uniformly  from rest to 6 m s-1  for the first 5 seconds,  then remained at a constant 6 m s-1  for the next to seconds then decelerated (negative acceleration/slowed down)  at a constant rate to rest over the next 4 seconds.

In section A the skater is increasing speed from 0 to 6 m s-1 in a time of 5 s.

We can therefore work out an acceleration: acceleration = change in speed ÷ time taken

  = 6 m s-1 ÷ 5 s = 1. 2 m s-2

Notice a couple of things about the way I have done this.

• I have used units for all my quantities throughout the whole calculation

• i have written the units correctly (a space between the number and the units, a space between consecutive units such as metre and second,; negative index notation.

• If you were asked to describe the motion in NCEA you need to specify constant acceleration.

For section B, the gradient is zero so the acceleration is zero.

For section C, the final speed is zero and the starting speed is 6 m s-1  so the change in speed is (0-6)  = -6 m s-1 

The negative sign in this case indicates that the direction of acceleration is OPPOSITE to the direction of movement, This means that it is decelerating i,e, slowing down.

The amount of deceleration is -6 m s-1 / 4 s = -1.5 m s-2 .  Because the sign is negative, this means that the net force causing the acceleration must be backwards (opposite to the direction of movement). For example, this could be caused by the skater going from concrete to grass and hugely increasing the friction force.

Working out distance traveled on a speed-time graph

Distance on a speed time graph is given by the area under the graph. Let's go back to the graph above and look at section A. I have shaded it blue on the diagram below.