# Forces

## What we are learning: (component knowledge)

**Scalar and vector quantities:**

Scalar quantities have a magnitude, this simply says how big it is. Examples include mass, energy, temperature distance and speed. They all have a number and units associated with them.

Vector quantities have both a magnitude and a direction. Examples of these include all forces, displacement, velocity, acceleration and momentum.

The main point to see is with speed and velocity. They may both have a magnitude of 8 m/s, however, velocity will also have a direction. E.g. The bike had a speed of 12 m/s but it had a velocity of 12 m/s east.

Likewise, distance and displacement. The bike travelled a distance of 850 m but had a displacement of 850 m east.

**Contact and non-contact forces:**

Contact forces are forces which are in contact with the body that they are applying the force to. If you are pushing a shopping trolley around a shop, you are applying the normal contact force to it through the handle. Example of contact forces are: friction, tension and air resistance (drag).

In a non-contact force, there is no physical contact between particles. Examples of these are: gravitational force (weight), magnetic force (attraction or repulsion) and electrostatic force (attraction or repulsion).

**Gravity, weight and gravitational field strength:**

Gravity causes a force of attraction between objects with mass. The more mass an object has, the stronger the force is. We feel this force that gravity causes on Earth is weight. The gravitational field around Earth causes the force, weight, that pulls you towards the surface. If the person has more mass, the weight is bigger. If the planet had more mass, the weight is bigger.

We can calculate the size of this force using the formula: weight = mass x gravitational field strength of W = mg. The gravitational field strength is measured in N/kg and on Earth is is 9.8 N/kg. On different planets, the gravitational field strength is different. This means that although your mass is the same on Earth as it is on Mars, on Jupiter and even in the depths of space, your weight will vary wildly.

Don't forget that weight is a vector quantity so it will also have a direction. When you are on Earth, the direction of your weight is always towards the centre of the Earth.

**Resultant forces and vector diagrams:**

When there is more that one force acting on a body, you can cancel them out and replace them with one overall force, This is called the __resultant force__. As an example, imagine that there is a a boy pushing a trolley northwards (forces are vector quantities so have a direction) with a force of 15 N. Another boy is pushing on the same trolley with a force of 13 N in a southwards direction. The resultant force will be 3 N in a northwards direction.

A vector diagram allows us to plot accurately the forces on paper. Their magnitude is displayed as the length of the lines and their direction follows the direction of a compass. When plotted accurately, we can use the "missing" line as our resultant force.

**Work done and calculating the energy transfer:**

You need to know that the work done in moving an object is equal to the size of the force applied multiplied by the distance the object moves in the direction of that applied force. The formula is given as W = F x s

Note that instead of distance (scalar) we are using displacement S (vector). Force is in Newtons, displacement in Metres, and work done is in joules J. Obviously as the force applied increases or as the distance travelled increases, so will the work done or the energy used in doing the work.

Using this equation, we can give the unit of a joule a definition. If you push an item with the force of 1 N, for 1 m, then you will have used one joule of energy or 1 joule of work is done.

**Forces, elasticity, springs, the energy stored in them and types of deformation:**

If a force is applied to an object, you will see in later lessons that it will affect its motion. If, however, the object's motion is not changed, it can cause the object's shape to be changed. It can be stretched, crushed or twisted. If the object changes shape but returns to its original shape when the force is removed, it is said to be __elastic deformation__, such as gently pulling on a spring. If it permanently changes the shape, then it is said to me __inelastic deformation__, such as stretching the handles of a plastic carrier bag.

When you add a mass to the end of a dangling spring, the weight (force) will pull the spring down. This causes the spring to extend. __Hooke's law__ shows that as the force applied increases, so does the extension, these are proportional right up to the __limit of proportionality__. Once this point has been passed, this relationship stops as we have moved into inelastic deformation. The relationship is connected by a __spring constant__ which is specific to that spring. The formula is F=ke where k is the spring constant in N/m and e is the extension in m and the force is in N. Remember that this is the extension which is the extended length of the spring - original length of the spring.

When a spring is stretched, work has to be done to do it. We can link this to previous equations as we know that we add a weight (our force) and there is a movement (extenstion). We can then work out the amount of elastic potential energy stored in in our spring but we need to involve the spring constant as different springs will have different amount os energy sotred too. E_{e} = ½ k e^{2}. Elastic potential energy = 0.5 x spring constant x extension^{2}.

Note how similar this looks to the equation for kinetic energy: K_{e} = ½ m v^{2}

**Describing motion, distance and displacement:**

This is a fairly simple concept, however, it is easy to get tripped up by it if you are not aware that distance is a scalar quantity and displacement is a vector. If you walk around a 400m running track and return to the start, your distance travelled will be 400m. As you have returned to the start, however, your displacement will be 0m.

Distance can be in straight lines, curves bends, zigzags etc, however displacement is a straight line from the starting point to the final point and the direction of travel.

Examples are: Distance = 27m, Displacement = 44m North west.

**Speed and velocity:**

Just like with distance and displacement, speed is a scalar quantity and velocity is a vector. Speed = distance/time whereas velocity = displacement/time. Both of these are measured in m/s but velocity has a direction added to it. Linking back to the running track analogy above, remember that velocity is in a straight line. The best thing to point out is that the Moon orbits earth at a speed of just over 1000m/s. The velocity of the Moon is 0m/s because its displacement will be 0m as it returns to the original point.

**Acceleration:**

Acceleration is a measure of the rate of change of velocity. This means that acceleration is also a vector quantity. Acceleration, or average acceleration, is how much the velocity of an object changes, divided by the time it took to change.

a = Δv/t

Acceleration is measured in m/s^{2}. If an object is slowing down, then it has a negative acceleration or is decelerating.

You can also this equation to find out the missing one from the following: initial velocity (u), final velocity (v), acceleration (a) and displacement (s)

v^{2} - u^{2} = 2 a s

**Displacement-time and velocity-time graphs:**

It is important with both of these types of graph to remember that time (s) runs along the x-axis. In a displacement-time graph, you can calculate the velocity by the gradient of the graph. The steeper the line, the bigger the velocity and if the line is flat (zero gradient) then the object has stopped. (If it is using distance instead of displacement, it is no longer a vector and you would get the speed rather than the velocity from the gradient).

In a velocity time graph, the gradient gives you the acceleration, if the velocity is in m/s and the time in s, then the acceleration is in m/s^{2}. The steeper the gradient, the higher the acceleration and if it is a negative gradient, the object is decelerating. If the line is flat (zero gradient), the object is travelling at a constant velocity. Finally, the displacement (distance travelled) can be calculated by working out the area under the graph.

** Falling and terminal velocity:**

When a body falls, we are interested only in the forces acting vertically up and down. Weight acts downwards (caused by gravity) and drag (also called air resistance) acting in the opposite direction. When a person jumps out of an aeroplane, they have not begun to fall so the only force acting on them in a vertical line is their weight, this causes them to accelerate downwards. As they gain velocity towards the ground, drag will increase in an upwards direction. When drag grows to be equal and opposite to the person’s weight and the forces are balanced, they reach __terminal velocity__. At this point, they are no longer accelerating and there is no resultant force.

When the person opens their parachute, the drag suddenly increases which unbalances the forces so they have a negative acceleration towards the ground (or they begin to decelerate) and the velocity reduces until the forces once again become balanced and a new, slower terminal velocity is reached. Many people think that when you open the parachute, you “shoot” back upwards but you simply slow down quickly. These forces, whether balanced or unbalanced can be shown on free body diagrams.

**Newton's 1st law:**

When you see the free body diagram for an object, if the resultant force is equal to zero, then the object will either carry on moving at its current velocity (both speed and direction) or if it wasn’t moving before, then it will remain still. We saw a part of this in the last component, when a person is falling, as long as their weight and drag are equal and opposite (balanced) they will continue to move at the same velocity (terminal velocity).

**Newton's 2nd law:**

As the force you apply to an object gets bigger, so does the object’s acceleration. This is one of easiest to understand concepts in all of physics. If you push a shopping trolley with a small force, it accelerates a small amount. If you shove the trolley with all of your strength, it will accelerate much more.

This can have values attached to it using the formula F = m a

or

Force (N) = mass (kg) x acceleration (m/s^{2})

**Newton's 3rd law:**

When objects interact, their forces are equal and opposite. You may have hear that “for every action, there is an equal and opposite reaction”. The best way to describe this is through a couple of examples, firstly if you run along the street, your shoe is pushing backwards along the pavement. At the same time, friction between your shoe and the pavement are pushing in the opposite direction which will propel you forward. Just to prove that point, imagine doing it on an icy morning, you still apply the force backwards but there is no reaction force because the ice prevents adequate friction and you fall or flail about like a cartoon character.

When a cannon fires a cannon ball, the explosion within forces the ball in one direction and the reaction force causes the cannon to move backwards in the opposite direction. Incidentally, referring back to the previous component, F = m a, the forces are equal and opposite (F) but the mass of the cannon ball is much smaller so it will accelerate much more than then cannon. The very large mass of the cannon means that it will only have a small acceleration (recoil).

**Deceleration and the forces & energy transfers involved in braking:**

When a car slows down, there has been a force applied to slow it down, this has come through the brakes. We can even work out the force applied through the brakes using F = m a. Remember that acceleration is a vector quantity so a negative acceleration is our deceleration. Energy is transferred in this process, the kinetic energy of the car has to be transferred to another energy store and that is the thermal energy store meaning that the brake pads, brake discs and tyres will become hotter. In fact, sometimes, a vehicle can suffer from “brake fade” which is when they become so hot, they no longer work efficiently.

The faster a vehicle is travelling, the longer it will take to slow down and the heavier it is, the longer it will take to slow down. These both, once again are linked to F = m a

**Thinking, stopping and overall stopping distances:**

Before we start, don’t fall for the mistake of replacing “distance” for “time”. We are only interested in the distance covered by a vehicle during these processes.

The __thinking distance__ is the distance covered by the vehicle between the driver seeing the hazard and applying the brakes. This is very closely linked to their __reaction time__ which is normally between 0.2 s and 0.9 s. If the driver was using a mobile phone, taking drugs, under the influence of alcohol or even trying to sort out an argument between children on the back seat, their reaction time is increased and so the thinking distance is also increased.

The __braking distance__ is the distance travelled by the vehicle while the brakes are applied. This distance can increase if the tyres are worn or incorrectly inflated, the road is wet, icy or muddy, the vehicle has an excessive load/too much mass (linked again to F = m a or if the brakes are in need to repair.

Finally, we add these two distances together to produce the __overall stopping distance__. You need to be able to analyse data on these numbers and offer explanations as to why one of the two component distances has increased.

**Momentum (HT):**

Inertia is the tendency of an object to carry on doing what it is currently doing. That is either moving or not moving. The __inertial mass__ is a measure of how difficult it is to change an object’s motion. A falling feather has a very low inertial mass and a fully laden container ship at maximum velocity will have a very high inertial mass. It can also be defined as the force/mass ratio.

__Momentum__ is a property that all moving objects have, it can be calculated by multiplying the mass (kg) by the velocity (m/s). The units of momentum are kgm/s. The formula is p = m v

This is used to explain why very heavy objects moving slowly can be more difficult to stop than very small objects moving quickly. You can see that a 1 kg block moving at 20 m/s will have the same momentum as a 20 kg block moving at 1 m/s. Also note that momentum is a vector quantity so direction is very important.

**The conservation of momentum (HT):**

In a closed system (where energy does not enter or leave), momentum is conserved. This means that if a snooker ball hits another ball and stops instantly, the new ball will carry on with the exact same momentum as the first as 100% of the initial momentum is passed onto the second. Likewise in a collision, a ball moving to the left with p = 1.5 kgm/s hits a ball moving to the right with p = 1.5 kgm/s, the overall momentum before = 0 kgm/s so after the collision it must also equal 0 kgm/s. The balls may both stop or they may bounce off each other, either way, the overall momentum must remain the same. This is the __conservation of momentum__

In an __explosion__ such as in a canon firing, the momentum before = 0 kgm/s so it must also =0 kgm/s after. This means that if the ball leaving the cannon has a momentum of 250 kgm/s to the East, then the cannon must have a momentum of 250 kgm/s to the West so that they cancel out. Note the link to the above component about Newton’s 3rd law of motion and as momentum is a vector quantity, you can apply the logic of a free body diagram to cancel out values.

## Key words/terms for this topic

Acceleration Braking distance Centro of mass Conservation of Momentum Deceleration Deformation Displacement (distance) Elastic Elastic Deformation Equilibrium Gravitational Gravitational field strength Inelastic Inertia Inertial mass Limit of Proportionality Momentum Newton's second law Reaction time Recoil Resultant force Scalar Stopping distance Terminal Velocity Thinking distance Vector Velocity Weight

## Curriculum Health Check:

Q: What is the overall distance between a driver spotting a hazard and the vehicle coming to a complete stop?

A: The reaction distance

B: The stopping distance

C: The braking distance

D: The thinking distance

## What you need to know

Scaler quantities only have a magnitude (the dog ran at a speed of 5m/s) whereas vector quantities have both a magnitude and a direction (the dog ran at 5 m/s in a westerly direction). When drawing vectors, we tend to draw and arrow (direction) and the length of the arrow depicts the magnitude. Distance is a scalar quantity, displacement is a vector quantity, it has distance (magnitude) and a direction.

Forces are vector quantities and they can be either contact or non-contact. Contact forces physically touch (friction) whereas non-contact have no touching (gravitational attraction or magnetic forces).

Weight is a vector quantity as you have a weight (magnitude) and it always pulls towards the centre of the earth (direction). Weight can be calculated using your mass and the gravitational field strength where you are (on Earth it is 9.8 N/kg). The mass and weight of an object appears to all act from one point, this is the centre of mass.

Weight (N) = Mass (kg) x Gravitational field strength (N/kg) or W=mg

When more than one force are acting on a body, they can partially or totally cancel each other out. The overall force left over is the resultant force. You need to calculate these, e.g. 5 N pulling left and 8 N pulling right gives a resultant force of 8-5= 3N pulling to the right.

Work done is the energy transferred when moving an object, it can be calculated using this formula:

Work done (J) = Force (N) x Distance (m) or W=Fs

You need to describe these energy changes. If there is more friction between the surface the object is on and the object, the force will need to be higher so the work done will be higher. The extra energy will be transferred to heat because of the extra friction.

When forces are applied to objects, they do not all move, some are forced to distort, bend, compress or stretch. Extension of a spring is directly proportional to the force, if the force doubles, the length of the extension doubles. As long as the spring has not passed its elastic limit (returns to original size/shape when the force is removed), we can measure how much it will extend per Newton based on a spring constant:

You need to be able to describe the relationships between the three components of this equation and interpret data and graphs. You can also calculate the energy stored in a stretched spring using this equation:

Elastic potential energy (J) = ½Spring constant (N/m) x extension^{2} (m)

E_{e} = ½ x k x e^{2}

As speed uses distance to calculate it, speed is a scalar quantity. Speeds are almost always changing in cars, bikes or even us walking. It is useful to have a rough idea of speeds, you walk between 1 and 2 m/s and may run up to 3 m/s and on you bike, you may reach 6 m/s. The speed limit for cars in the town is 30 mph which equates to 13.4 m/s whereas the soundwaves coming off a car which allows you to hear it travels at about 330 m/s.

You need to calculate speed, distance and time by using the equation below and rearranging it as needed.

Distance travelled (m) = Speed (m/s) x Time (s) or S = vt

Velocity is a vector quantity so although you can calculate it the same as speed, it must have a single direction.

Using a graph of distance (y-axis) against time (x-axis), you can calculate the speed of an object as the gradient of the line.

You can calculate the average acceleration of an object using this formula:

Acceleration (m/s^{2}) = change in velocity (m/s) / time taken (s)

Another way to calculate this is using a velocity-time graph. The gradient of this graph is equal to the acceleration. You need to be able to draw velocity-time graphs from data or interpret the graphs to obtain data.

For uniform acceleration, you can use the following equation;

Final velocity^{2} (m/s) - Initial velocity^{2} (m/s) = 2 x acceleration (m/s^{2}) x distance (m)

v^{2} - u^{2} = 2 a s

The acceleration due to gravity on Earth is 9.8 m/s^{2}. This is an example that you could use to calculate the final velocity of an object falling to the ground. In reality, the air resistance acting on a falling object increases as its velocity increases. At some point, the air resistance (N) will equal the weight of the falling object (N) and they will cancel out giving a resultant force of 0 N. At this point, there will be no more acceleration.

Newton's 1st law of motion. If the resultant force on a body is zero, a stationary body will remain stationary whereas a moving body will continue to move at the same velocity. A resultant force is needed to change velocity.

Newton's 2nd law of motion. The acceleration of an object is proportional to the resultant force. Bigger force = bigger acceleration. This is calculated using this equations:

Force (N) = Mass (kg) x Acceleration (m/s^{2}) or F=ma

Required practical 19 - Investigate how varying the force on an object affects the acceleration when the object's mass is constant. See how the acceleration is affected by keeping the force the same but varying the mass of the object.

Newton's 3rd law. When two objects interact, they exert equal and opposite forces on each other. You need to be able to apply this to equilibrium situations.

Overall stopping distance for a car is equal to the thinking distance (distance covered between seeing the hazard and applying the brake) plus the braking distance (the distance covered while the brake is applied until the car comes to a stop). Please note that I have been very careful to state "distance", don't quote times - just distances. The faster the car, the longer the overall stopping distance.

Thinking distance is affected by reaction time. Your braking distance will increase due to tiredness, use of drugs or alcohol.

Braking distance will increase if the car's tyres have low tread or the air pressure is at an incorrect level, the road is wet, icy, covered in mud or has loose gravel. Be prepared to calculate distances based on equations above and link to acceleration and then to braking force.

Larger decelerations need a greater force and these rapid decelerations are dangerous as brakes can overheat and become much less effective or the car can lose control.

**Extra topics needed for the Higher Tier papers:**

You should be able to describe the forces acting on a stationary body and use free body diagrams to show how several forces can lead to one resultant force.

Single forces can be split into two component forces that add up (taking vector directions into account) the single force. You need to resolve forces using vector diagrams.

If a dog runs around in a circle at 4 m/s, its speed is 4 m/s but its velocity is 0 m/s. This is because the direction is constantly changing so there is no velocity.

On a distance-time graph, if an object is accelerating, the line becomes a curve. You can work out the speed at any point on the curve by drawing a tangent and working out the gradient of this tangent.

On a velocity-time graph, the area under the graph is equal to the displacement of the object. You may need to use a graph to calculate the distance travelled either by shapes (splitting the area into squares and triangles) or counting squares of the graph paper.

Inertia is the tendency of a body to either stay at rest or have a constant velocity. It resists change. Inertial mass is a measure of how difficult it is to change an object's velocity and is defined as the ratio of force over acceleration.

When accelerating, the force from the engine can be calculated using F=ma, likewise, once the deceleration is known, you can use F=ma to calculate the braking force.

The momentum of an object can be calculated using this formula:

Momentum (kgm/s) = Mass (kg) x Velocity (m/s) or p = mv

In a closed system or "an explosion" scenario, the momentum before an event is equal to the momentum after. This is crucial as momentum uses velocity to calculate it and as velocity is a vector quantity, so is momentum.

This page was updated on: 26th March 2024