Interactive text book: Forces
What makes things move?
So far we have been dealing with movement. We can describe motion in terms of displacement, velocity and acceleration and solve problems using the 'suvat' equations provided acceleration is constant.
But what makes things move? If we just had the observations of stars and planets to go on this would be a very difficult question to answer. We need to look in the lab and in our imaginations because we cannot observe completely isolated bodies. If we could, we would notice that isolated bodies are either at rest or moving with constant velocity unless they hit each other, then they change direction and speed. We can simulate this situation in Algodoo.
When the red ball hits the blue one they push each other. The blue ball pushes the red one to the right and the red one pushes the blue one to the left. We say they have exerted a force on each other. For there to be a force there must be two bodies, one to experience the force and the other to exert it.
Newton's third law
If body A exerts a force on body B then body B exerts an equal and opposite force on body A.
In a Newton's third law pair, the forces:
- Act on different bodies
- Are the same type
- Are equal in magnitude and opposite direction
In the animation above the magnitude of the force exerted on the blue ball is:
The force is equal and opposite.
Balanced forces
A force is a push or a pull. We have seen that a force applied to a body changes the velocity of the body. However, if two equal forces act on the same body in opposite directions, the body doesn't change its velocity.
We say the forces on the green ball are balanced.
In the animation above the force on the red ball equals:
All the forces between balls have equal but different directions.
Newton's first law
A body will remain at rest or in motion with constant velocity unless acted upon by an unbalanced force.
Mass
In the previous two animations all the balls were all identical. What if they are different?
The force experienced by each ball is the same but the red one has a bigger change of velocity. The time of the collision is the same for each so we can deduce that the acceleration of the red ball is greater than for the blue one. What causes the blue ball to have a smaller acceleration? Is it its size? If we watched lots of different balls colliding we would see that it's nothing to do with size.
It's the mass that is important. We call this inertial mass. Mass is the property of an object that resists acceleration.
This is confused somewhat by the unit of mass, the kilogram, which was in fact originally defined by an amount of matter, the mass of a litre of water at \(0 \text{ ℃}\). As a result, some IB physicists mistakenly believe that mass is defined as the quantity of stuff in a body.
In the animation above the density of red ball is:
The mass of the red ball is greater but it has a smaller volume so it is more dense.
Newton's second law
The acceleration of a body is directly proportional to the unbalanced force and inversely proportional to its mass:
\(a\propto {F\over m}\)
Note that the unit of force, the Newton, is defined so that if \(a\) is measured in \(\text{m s}^{-2}\) and \(m\) in \(\text{kg}\) the constant of proportionality is \(1\):
\(a={F\over m}\)
So \(1\text{ N}\) is the unbalanced force that would cause \(1 \text{ kg}\) of water to accelerate at \(1\text{ m s}^{-2}\). This would have been quite difficult since water is a liquid but they since changed the definition of the kilogram to be the mass of a special lump of iron, the international prototype of the kilogram.
A 2 kg body accelerates at 5 m s-2.
The unbalanced force on the body is:
F = ma
Force as a vector
We will come back to Newton's laws of motion later but now we will look a bit more closely at forces. We have seen that an unbalanced force causes a body to accelerate and that the direction of acceleration depends on the direction of the unbalanced force. Force is a vector.
During the collision:
- the blue ball has negative acceleration so the force is to the left.
- the red ball has positive acceleration so the force is to the right.
When adding forces we must use vector addition. In this example the resultant force is \(2.00\text{ N}\) to the right:
In the next example the forces are perpendicular so we can find the resultant using Pythagoras:
\(F=\sqrt{3^2+4^2}=5\text{ N}\)
We can also take components of a force:
The vertical component is \(F\sin\theta\) and the horizontal component is \(F\cos\theta\).
This is very useful when finding the resultant of non-perpendicular forces. Instead of using complicated trigonometry we find the horizontal and vertical components of all forces and then add these then using Pythagoras to give the resultant. Here is an example:
The vertical components are \(2\times 5\sin 37=3+3=6\text{ N}\).
The horizontal components cancel: \(5\cos37-5\cos37=0\text{ N}\)
The resultant force is therefore \(6\text{ N}\) vertically.
If F = 6 N and θ = 30° the vertical component of F is:
Vertical comp. = 6 x sin30
Types of force
In the conditions for a Newton's third law pair, we said that forces should be the same type. Let's look at different types of forces.
Contact (normal) forces
In the example of colliding balls the only way one ball can exert a force on another is by colliding. This is called the contact force or normal force. The reason it is called the normal force is because it is perpendicular to the surfaces in contact. We can see this in the Algodoo simulation.
Tension
It is also possible to exert a force without the bodies colliding. For example, the bodies could be connected with a rope. This force is called tension.
You can see that the tension acts on both bodies, to the right on the red one and to the left on the blue one (with the same magnitude in each direction).
Gravitational force (weight)
This is a weird one! When we release a ball in the lab it falls to the ground without anything touching it. This may not seem weird to you because you are used to this happening. The reason the ball falls is gravity... but what is gravity? It's the thing that makes the ball fall. Not entirely convincing but that's the way it is. If we take our lab into deep space the ball will not fall, so it's something to do with the Earth. Newton looked at the numbers and came up with his universal law of gravity:
Every particle of mass attracts every other particle with a force that is directly proportional to the product of their mass and inversely proportional to their separation squared.
\(F_\text{G}={GMm\over r^2}\)
The constant of proportionality, \(G\) is very small, so the force between small masses is negligible. The Earth has a large mass (\(6\times10^{24}\text{ kg}\)) so the force of gravity on \(1\text{ kg}\) placed close to the surface of the Earth turns out to be \(9.8\text{ N}\). You may notice that this is the same as the acceleration of the ball - stranger still! It took Einstein to get to come up with a theory to properly explain this through the theory of general relativity, but we don't need to go into that here.
Because everything in our lab is close to the surface of the Earth we can assume that the force pulling all objects down is proportional to their mass, the constant of proportionality is \(g = 9.81\text{ N kg}^{-1}\):
\(F_\text{G}=mg\)
If the gravitational force is the only force acting on a body it will have acceleration \(a\). From Newton's second law we know this equals \(F\over m\).
\(F=ma=mg\)
\(\Rightarrow a=g\)
So we can define mass in two ways:
- The inertial mass that resists acceleration
- The gravitational mass that is responsible for gravity
Two seemingly different things that have the same size. This is very convenient as it means we can find the mass of a bag of sugar by measuring its weight instead of having to throw it across the room!
What we need to know when dealing with gravitational force (weight) is that it acts vertically downwards on the centre of the body (the centre of gravity) and equals \(mg\).
Don't forget though that there must always be two bodies. It is the Earth that is exerting this force so, according to Newton's 3rd law, there must be an equal and opposite force on the centre of the Earth.
Balancing forces
Now we have three different forces to play with we can think about some examples where combinations of forces act on a body.
First, let's consider two balls of different mass hanging off the ends of a string that is running over two pulleys. The heaviest ball rests on the ground and the lighter one hangs in the air.
Notice that in this diagram we have only included the forces acting on the bodies of interest and not the forces exerted by the body. This is called a free body diagram.
Both balls are at rest so Newton's first law implies that the forces are balanced. For the black ball:
\(T=mg\)
For the green one:
\(T+N=mg\)
The next example involves a box on a frictionless slope prevented from sliding down the slope by a rope.
The box is at rest so the forces are balanced. If we add the vectors they will form a closed triangle.
This also means that the components of the forces in any direction are balanced. In this example it is more useful to take components parallel and perpendicular to the slope rather than horizontally and vertically.
.png)
Parallel to the slope: \(mg\sin\theta=T\)
Perpendicular to the slope: \(mg\cos\theta=N\)
Combining these by dividing the parallel equation by the perpendicular equation:
\({mg\sin\theta\over mg\cos\theta}=\tan\theta={T\over N}\)
A ball on a string rests against a wall.
Which of the following are equal and opposite?
The body is at rest so horizontal forces are balanced.
Friction
In reality you might not need a rope to stop the box sliding down the slope because there is friction between the two surfaces. Friction opposes the relative motion between two surfaces that are in contact. We can see the frictional force in the following animation.
The friction opposes the force pushing the box. As this force increases so does the friction but only up to a point. At \(43\text{ N}\) the box starts to slip. This point depends on the coefficient of friction for the surfaces.
Static friction is the force that prevents a body from starting to move. This equals the force pushing the body but has a maximum value (in the example above it was about \(43\text{ N}\)). If the pushing force exceeds this value the body starts to accelerate. The maximum value depends upon the type of surface and the normal force between the surfaces:
\(F_\text{f, max}=\mu_\text{s}N\)
If the rope is cut but the box remains at rest then the friction must be equal to the component of weight acting down the slope.
Parallel to the slope: \(F_\text{f}=mg\sin\theta\)
If this is the maximum angle before the box slides then:
\(F_\text{f}=\mu_\text{s}N\)
\(N=mg\cos\theta\)
\(F_\text{f}=mg\sin\theta=\mu_{s}mg\cos\theta\)
\(\mu_\text{s}={\sin\theta\over\cos\theta}=\tan\theta\)
A simple way of measuring μs is to determine the maximum angle of slope.
The maximum angle is \(45°\) so \(μ_\text{s} = \tan45° = 0.5\).
Dynamic friction is the force that opposes the motion of two surfaces that are in contact. It is a fixed value that depends on the nature of the surfaces and the normal force but is independent of the velocity:
\(F_\text{f}=\mu_\text{d}N\)
The coefficient of dynamic friction is less than the coefficient of static friction (although in Algodoo they are the same!).
Note that friction doesn't always slow things down; it also allows then to move. If you try to drive a car on a friction-free road it wouldn't go forwards as the wheels would just spin. The lower surface of the spinning tyre moves backwards relative to the car. Friction acts against this motion pushing the car forwards.
A man attempts to push a large box across the floor but fails.
If he exerts a horizontal force of 500 N on the box the friction between the box and the floor
The forces are balanced.
Air resistance (drag)
Air resistance is the force that opposes the motion of a body through the air (or any other fluid).
There are different models of air resistance. For slow moving spheres the air flows around the body causing a force that is proportional to velocity.
\(F=6\pi\eta vr\)
\(η\) is a constant for the fluid called viscosity and \(r\) is the radius. This is Stokes' law.
For faster moving objects like cars the air is pushed aside by the front surface of the body. This force is proportional to \(v^2\).
\(F={1\over 2}A\rho Cv^2\)
\(C\) is a constant called the drag coefficient and \(A\) is the area of the front of the body.
Since the drag on a car is proportional to \(v^2\), drag will increase as the car goes faster. At some point drag will equal the forward force created by the engine. When the forces on the car are balanced, the car will not accelerate. It has reached its top speed. For a falling object, we call this terminal velocity.
To reduce drag, the area of the front surface could be reduced...
...but unfortunately makes the car rather difficult to get into!
To double the speed of a car the force exerted by the engine must be increased by a factor:
Drag = \(F={1\over 2}A\rho Cv^2\)
at const. vel. Engine force = drag
2v results in 4 x drag
Buoyancy
Buoyancy is another force caused by the surrounding air or fluid and one that is acting on you right now. It is an upward force equal to the weight of fluid displaced.
If we put a metal ball into a completely full container of water an amount of water equal to the volume of the ball will overflow. This water has been displaced by the ball. The ball will experience an upward force equal to the weight of this water (according to Archimedes' principle).
A body floats if the volume of fluid equal to the weight of the body is less than the volume of the body. This means the density of the fluid needs to be greater than the density of the body. In the example below the density of the fluid is twice the density of the ball.
Even though only half of the ball is submerged the buoyant force is equal to the weight of the ball.
Helium is less dense than air so the air it displaces has a greater weight than the balloon. This causes helium balloons to accelerate upwards. As the velocity increases the drag on the balloon also increases until the buoyant force equals the sum of the weight and drag. The forces are now balanced so the velocity remains constant.
A metal ball hanging on a string is immersed in water.
The string is cut.
Immediately after the string is cut:
Immediately after the string is cut the ball is not moving
A metal ball hanging on a string is immersed in water.
The string is cut.
After 0.1 s the ball is accelerating downwards
The forces are unbalanced
A metal ball hanging on a string is immersed in water.
The string is cut.
After 2s the ball reaches terminal velocity:
Immediately after the string is cut the ball is not moving
A metal ball hanging on a string is immersed in water.
The string is cut.
After 3 s the ball rests on the bottom:
Weight = Buoyancy + Normal force
Free body diagrams
Throughout this section only the forces acting on the body of interest have been drawn. All the forces exerted by the body have been left out. This is called a free body diagram.
Using free body diagrams make it possible to find the resultant force on a body. It's also best to draw only original forces and not their perpendicular components to avoid a very confusing picture!
