2. MECHANICS

2.1. Mechanics - the foundation of physics

The first and most important part of the areas of physics is mechanics, which forms a basis for other parts to be presented later. The quantities used here will reappear in many places - there are many types of forces, but all follow the laws of Newton. Velocity is not only a quantity to be studied for its own sake, but for example the velocity of an electric charge affects how it reacts to magnetic fields. The mechanics studied in this topic is classical mechanics, developed mainly in the time period 1600-1800. A more precise modern theory of mechanics, involving Einstein's theory of relativity and quantum mechanics can be learned later. For most technical applications - including advanced technology like sending a spacecraft to the planet Mars - this classical mechanics is still sufficient.

2.2. Distance and displacement

We start the physics course with mechanics which deals with questions like where something is, how fast and in what direction it moves, how its motion changes, what causes it, and some consequences of the answers to these questions. All this fits the universal character of physics. An example of this is the quantity speed (described later): a car may drive at a speed, an animal may run or fly at a speed, blood can flow through your veins at a speed, a distant star or galaxy may move towards or away from us at some speed.

Before we get to the quantity speed, we need to describe something more fundamental: where something is. In physics, there are two ways to tell how far something has moved or how far from a certain point it is:

distance = how you went measured along the path you actually took. The tripmeter in a car measures distance. Since the road can be curved it is difficult to say what direction you took, and distance is then a scalar. Common symbol : s

displacement = how far it is from where you started to where you stopped in a straight line. If you look at the map maybe you can find out that the town you drove to is 25 km to the northwest of where you started. This is a vector, which also often has the symbol s.

If only two directions are possible, it is convenient to use positive values for displacements in one direction and negative in the opposite.

Ex. The train moved 500 m forwards and then 200 m backwards.

* If we call the forwards direction positive, the total displacement is 500m + (-200m) = 500 m - 200 m = 300 m.

* If we call the backwards direction positive, we have - 500 m + 200 m = - 300 m.

Note:

·        in both cases we could add the displacements, with their signs. The same formula could have been used: s_total = s₁ + s₂

·        the answers are different although they represent the same motion in the real world. They must be interpreted using the chosen definition of which direction that is positive.

2.3. Speed and velocity

This can also be described either with or without a direction:

speed = distance/time This is a scalar. Unit ms^-1

velocity = displacement/time This is a vector, same unit.

(note that 1 ms^-1 = 3.6 kmh^-1)

The same formula is used for both (v for velocity or speed, s for distance or displacement, t for time):

v = Ds/Dt [DB p. 4]

The symbol D stands for the change in something = the difference between what it is now and what it was before. In many situations it can be dropped - for example the time for something to happen is the difference between what the clock showed after it and when it started, but if we started a stopwatch from zero when the event started, then the reading on the stopwatch when the event is over equals the time it took. We then often use the formula in the form

v = s / t

If the velocity is constant (both magnitude and direction!), we have what is called uniform motion, UM.

Frames of reference and relative velocity

Example: A boat A moves with 5 ms^-1 downstream, another boat B with 5 ms^-1 upstream in a river flowing 2 ms^-1 relative to the shore. Both move at 5 ms^-1 in the "river's frame of reference", but their speeds in the "shore frame of reference" (or their speeds relative to the shore) are 3 ms^-1 and 7 ms^-1.

2.4. Acceleration (acceleration)

If the velocity changes (magnitude and/or direction), we have an acceleration. We will first focus on cases where something moves along a straight line, but where the speed = the magnitude of the velocity changes. We use these symbols:

u = initial velocity

v = final velocity

t = time to change velocity from u to v

a = acceleration

Dv = v-u = change in velocity

The definition of acceleration is then

a = Dv/Dt [DB p. 4]

where we can write Dv/Dt = (v - u) / t (assuming that t = the time it took for the velocity to change from u to v). Acceleration is a vector and its unit is ms^-2 (which means m/s²). The formula is often written in this form after solving for v:

· a = (v - u) / t multiply both sides with t so

· at = (v - u) = v - u move - u to the left side, letting it change sign

· at + u = v or as below:

v = u + at [DB p. 5]

If the acceleration a is constant, we have uniformly accelerated motion, UAM.

·     Near earth, all things fall down with a gravity acceleration g = 9.81 ms^-2 if we do not think of air resistance.

• For UM we would have a = 0 and v = u + at would become v = u ; the velocity is constant

2.5. Graphs of UM and UAM

UM:

·        The graph of velocity as a function of time (velocity on y-axis, time on x-axis) is a horizontal straight line (the velocity is constant). If an object has travelled for the time t with the velocity v, the displacement (how far it as moved) is given by v = s/t => s = vt. This is the of the area (rectangle) under the graph.

·        The graph of displacement as a function of time is a straight line which is steeper the higher the velocity is. Compare this to the graphs of y = x, y = 2x, y = 3x etc where the graph y = kx is steeper the higher k is. Here we have s = vt with s instead of y, v instead of k and t instead of x. The velocity is now the gradient (slope) of the line. This means that you take any two points A and B on the curve and find how much higher B is than A, then divide it by how much further to the left B is than A.

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UAM:

·        The graph of v as a function of t is now a rising straight line starting from u (initial velocity) on the velocity axis. During the time t it reaches the level v (final velocity). The distance travelled is still the area under this graph - now a trapeze (like a triangle on top of a rectangle). This area can be found by adding the areas of the rectangle and the triangle or by finding the mean or average velocity v_m which is (u+v)/2. Since s = v_mt we then get:

s = [(u + v)/2]t [DB p. 5]

·        The graph of displacement s as a function of time is now not a straight line but a curve bending upwards (getting steeper and steeper - the gradient or slope is still = the velocity, but since this increases all the time, we would need to draw a "help line" (called tangent) and find the gradient = slope of this by choosing two points on it)

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Other types of motion (neither UM or UAM)

If the velocity is not constant (not UM) and the acceleration not constant (not UM) is still true that the travelled displacement is the area under the v-t curve (which may be found with geometry, numerical approximations on a computer, or other methods) and that the velocity at a certain time is the slope of the s-t curve.

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Instantaneous and average values

If one quantity is the gradient (slope) of another (e.g. velocity from displacement or acceleration from velocity) we can graphically find either an average or an instantaneous value. The average value is the change in the vertical coordinate / the change in the horizontal coordinate. The instantaneous value is the "average" value for an infinitely small change in the horizontal coordinate.

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2.6. The 4 equations of uniformly accelerated motion = UAM

We already have

v = u + at [DB p. 5]

and

s = v_mt where v_m = (u+v)/2 [DB p. 5]

We can now

·        replace v in 2) by u +at and get vm = (u+u+at)/2 = (2u + at)/2

·        simplify v_m = (2u +at)/2 = 2u/2 + at/2 = u + ½at

·        to get s = v_mt multiply with t and have t(u + ½at) = ut + ½at²

so we have:

s = ut + ½at² [DB p. 5]

Another possibility is to

·        solve 1) for t which gives t = (v - u)/a

·        replace t in 2) with this, so s =v_mt = v_m(v - u)/a

·        use from 2) that v_m = (u + v)/2 to get s = (u + v)(v - u)/2a

·        let (u +v)(v - u) = (v + u)(v - u) = vv - vu + uv - uu = v² -vu + vu - u² = v² - u²

which all gives us

·        s = (v² - u²)/2a which we multiply with 2a to get v² - u² = 2as

·        and finally v² = u² + 2as so :

v² = u² + 2as [DB p. 5]

Note that the equations are valid only for constant acceleration!

2.7. Force and mass

Force (a vector quantity)

is the cause of for example

· deformation (stretching, bending, compressing, other). It is measured with a forcemeter (dynamometer, newtonmeter) containing a spiral metal spring which is extended (stretched out) more the greater the force is. Unit : 1 newton = 1 N.

· acceleration = change in velocity per time.

Resultant force (resultant, total force, net force, sum of all forces)

Often several forces act on the same object. If you hold something in your hand, there is a force of gravity pulling it down and a force from your hand upwards which may balance out the downwards force so the resultant is zero. This can be handled by choosing one direction as positive (ex. up) and giving the forces signs accordingly.

Example:

Force of gravity = F_g = - 5.0 N Force of hand = F_h = 5.0 N

Resultant = F_g + F_h = - 5.0 N + 5.0 N = 0

Newton's 3 laws for forces :

Newton I

If the resultant force on an object is zero, its velocity will be constant.

This can mean either of two possibilities:

·        the object is at rest and will remain so as long as the resultant is zero (like the object in your hand).

·        the object has some velocity and will keep it (both direction and magnitude) as long as the resultant is zero. Example : A car comes to a curve where the road is slippery because of ice. The driver would like to either slow down or change direction, but because of the ice no force can be applied to it horizontally, so it continues out into the forest where Forces from trees it collides with slows it down. (This was in the horizontal dimension - in the vertical dimension there is a force of gravity down which is balanced out by the force from the hard ice in the road keeping it from sinking into it).

A free-body diagram = sketch of an object showing the forces acting on it using arrows with a length proportional to the magnitude (if known). Forces (as other vectors) can using trigonometry be resolved into components in two dimensions perpendicular to each other, and the components added separately. The resultant force/ resultant/ total force/ net force can be found using Pythagoras.

Translational equilibrium = a situation where the net force in all dimensions is zero. Example: an object sliding down a slope at constant speed, when the component of the force of gravity down the slope and the force (ex. friction) balance out, and the same is true for the normal force (perpendicular to surface) and the component of the force of gravity perpendicular to the surface (draw diagram, choose labels, resolve into components!).

Newton II

If there is a resultant force F, then there will be a change in velocity = acceleration

which is greater the greater F is, but smaller the greater the mass of the object is.

a = F/m

A larger engine giving a larger net force will increase acceleration

A larger mass will decrease it.

F = ma [DB p. 5]

This means that the unit 1 N = 1 kgms^-2. Mass is a scalar, but acceleration is a vector, so the force is also a vector.

Newton III

If A acts on B with the force F then B acts back on A with - F

(-F is a force of the same magnitude but opposite direction to F). Examples:

·        A rifle fires a bullet and acts with a force on it accelerating it forwards, but the bullet acts back on the rifle so it recoils

·        A rocket engine in a space ship throws out gases acting with them, and then the gases act back on the rocket with a force forwards (note that the rocket does not "push against the air" to drive it forwards, it works out in empty space).

Mass and weight

·        mass is a property of an object which it has wherever we take it - a 100 kg astronaut is a 100 kg astronaut here or on the moon

· weight is the force of gravity acting on something - on the moon where the force of gravity is weaker, the weight in Newton’s is lower.

The force of gravity is

F_g = mg

where g = the gravity constant or gravity acceleration = 9.81 ms^-2 on earth, 1.6 ms^-2 on the moon.

· Inertial mass = F/a (where F is resultant force, regardless of what kind of force - force of gravity, force of hand, force of rocket engine, electrical forces or other).

· Gravitational mass = F/g (near earth) the property of an object which determines how large the force of gravity on the object is.

There is basically no "good" reason why the inertial and gravitational masses should be the same - why the quantity which says how much force of any kind is needed to accelerate an object should be the same as the one which says how strong one particular force (gravity) is. For the three other fundamental forces (electromagnetic, strong and weak nuclear force) the strength of the force is determined by other quantities (ex. electric charge).

2.8. Work, energy, power

Work and energy

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If the force or a component F_s of it is in the direction of its displacement, the work (a scalar) done is

W = (F_ss =) Fs cosq [DB p. 5]

with the unit 1 joule = 1 J = 1 Nm. The amount of work done is the energy (same unit) converted from one form to another.

In a velocity-time diagram the displacement is the area under the graph since s =vt for UM, for other types of motion the area is not a rectangle but still equal to s. Similarly, in a graph of F_s as a function of s, the area under the graph - rectangle or other - is the work W.

Kinetic energy

·        if a car is accelerated from rest by the constant horizontal force F then the work done is W = Fs = mas; here q = 0

·        from the equation for UAM, v² = u² +2as we now get v² = 2as and then a = v²/2s

·        inserting this in W = mas, gives W = ½mv² which is "stored" in the moving car, so

E_k = ½mv² [DB p. 5]

Gravitational potential energy

·   if an object falls from the height h the force of gravity does a work W = Fs = mgs = mgh on it:

E_p = mgh [DB p. 5]

The sum of these is the total mechanical energy, which is constant (that is, conserved) unless energy is lost to do work against friction, air resistance or other.

Power

P (= E/t or W/t) = work/time = Fv [DB p. 5]

unit 1 watt = 1 W = 1 Js^-1. Power is the amount of work done or energy transformed from one form to another per time; it can be called the rate of working. "The rate of X" means "how much X per time". Note that for an object moving at a constant speed v the power P = W/t = Fs/t = Fv where F is not the resultant force but the force keeping it in motion despite friction, air resistance etc. Note the older unit 1 horsepower = ca 735 W.

Efficiency

e or h = E_out/E_in or P_out/P_in [not in DB but a similar definition is given in thermal physics, DB p.6]

where E_in is the work or energy supplied and E_out that which is converted to something "useful". What this is depends on the purpose of the device; for a light bulb where a certain amount of electric energy is supplied, the useful energy is that converted to light and the energy converted to heat wasted. For a bread toaster, it is the opposite. Power can be used instead of work or energy since the time t is cancelled: P_out/P_in = (E_out/t)/(E_in/t) = E_out/E_in

2.9. Friction

Friction

The force of friction is caused by interaction between atoms in the material of a surface and in an object in contact with it. For the force of friction we have

F_fr = m_kN and F_fr < or = m_sN [DB p. 5]

m = positive friction coefficient, without unit, which can be

· kinetic (index k) or dynamic or sliding for moving object (force opposite to velocity)

·        or static (index s) for object at rest (force opposite to net force trying to set it in motion). In this case the value is such that the force of friction balances any net force trying to set the object in motion until some maximum value, when the object "jumps" into motion and the force of friction then is kinetic (with a constant coefficient somewhat smaller than the maximum value of the static one)

N = normal force, the force with which the surface is pressing towards the object (on a horizontal surface N = -F_G so it can be replaced by the force of gravity in a calculation where only magnitudes are involved.

Alternatively: We use different positive-negative directions in the horizontal and vertical dimensions. This means that N or F_N (which is in the vertical dimension, balancing out the force of gravity G or F_G) may be given a different sign when used to calculate the force of friction as the expression mN since m is always positive and the force of friction can be either positive or negative depending on our choice of directions. The force of friction is, in principle, not affected by the area of the object which is in contact with the surface.

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For an object on an incline (slope) it must be noted that the normal force is not the opposite of the force of gravity, but of the component of the force of gravity perpendicular to the slope.

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For a moving object, F_fr is in the opposite direction to the velocity. For a static object, it is in the opposite direction to the resultant of all other forces acting on it. (Important!)

2.10. Springs

Linear springs

If a spring is extended (pulled out) or compressed (pushed in) a displacement x it acts with a force according to

F = (-) ks [DB p. 5]

A force which follows this type of a formula is called a harmonic force. 

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where k = spring constant, unit Nm^-1 (higher the k the stronger the spring is); the minus sign shows that the force of the spring is in the opposite direction to the displacement s from the equilibrium position

Elastic potential energy

When a spring is extended or compressed, work is done on it which can be stored in it as an elastic potential energy. Since the force needed to overcome the force of the spring is not constant but increases linearly the work done = the area under the force graph = ½ * the base * the height = ½ * x * F = ½ * x * kx =

E_elas = ½kx² [DB p. 5]

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2.11.* Simple harmonic motion

Mass on spring

It can be shown that for a mass m oscillating on a spring with the spring constant k, the time period T for the oscillations follow the formula:

T = 2pÖ(k/m) [not in DB]

Simple pendulum

In a similar way it can be shown that for a mass m (sometimes called the pendulum "bob") swinging at the end of an assumed massless pendulum of the length l has the time period

T = 2pÖ(l/g) [not in DB]

2.12. Momentum and impulse

(Linear) momentum

a vector quantity, unit 1 kgms^-1 , is defined as:

p =mv [DB p. 5]

If we define momentum p = mv we can also write NII as F = Dp/t (meaning "net force is the rate of change in the momentum") since initial momentum = mu, final momentum = mv and change in momentum per time = (mv - mu)/t = m(v - u)/t = ma = F. Note that momentum = Fi , all = torque, a quantity to be presented later.

Note: here F is the resultant force

F = Dp/Dt [DB p. 5]

When two objects A and B collide or otherwise interact for the time t and no external force is acting (e.g. the force of friction can is neglected), the total moment is conserved (the same before and after the collision) since

·        N III : A acts on B with F so B acts on A with - F

·        no external forces, so these are the resultant forces on A and B

·        N II for A: - F = ma_A = m(v_A - u_A)/t = (mv_A - mu_A)/t = Dp_A / t

·        N II for B: F = ma_B = m(v_B - u_B)/t = (mv_B - mu_B)/t= Dp_B / t

·        therefore Dp_A/t = - Dp_B/t and Dp_total = Dp_A + Dp_B = 0

·        no change in total momentum means it is the same before and after 

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In calculations for problems with two objects colliding, the most useful form of this is

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ [not in DB]

where the formula is adapted according to the situation, e.g. :

·        if object 2 was at rest before the collision then u₂ = 0 and the term m₂u₂ dropped

·        if the objects stay together after the collision, then v₁ = v₂ = v and m₁v₁ + m₂v₂ = (m₁ + m₂)v

·        one direction is chosen positive, and the velocities given positive or negative values accordingly. If a velocity is calculated, the sign shows its direction

Since momentum is a vector we can have collisions in two dimensions where the momentums and/or the velocities are split up into components in two perpendicular dimensions. These are then both conserved m₁u_1X + m₂u_2X = m₁v_1X + m₂v_2X and m₁u_1Y + m₂u_2Y = m₁v_1Y + m₂v_2Y). The components of the momentum are found using trigonometry like for velocities.

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Another useful relation is the following: Since p = mv => p² = m²v² => p²/2m = ½mv² so:

E_k = p² / 2m [DB p. 5]

Impulse

I = FDt = Dp [DB p.5]

(unit 1 kgms^-1 = 1 Ns) where F is the resultant force acting on an object, t the time during which the force acts (can be a very short time for a collision). If the Force acting is not constant, the only way to find the impulse and with that the change in momentum, is to find the area under the graph of F as a function t. If we find the impulse from the graph, then I = Dp = m(v-u).

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Elastic collisions

In an elastic collision, e.g. two hard billiard balls colliding and bouncing apart, the total kinetic energy is also conserved.

Example: A billiard ball A with the mass m and velocity u_A collides elastically with another identical ball B at rest. What will happen?

Conservation of momentum: mu_A + mu_B = mv_A + mv_B

=> mu_A = mv_A + mv_B

=> u_A = v_A + v_B

Conservation of kinetic energy: ½mu_A² + ½mu_B² = ½mv_A² + ½mv_B²

=> ½mu_A² = ½mv_A² + ½mv_B²

=> u_A² = v_A² + v_B²

=> (v_A + v_B)² = v_A² + v_B²

=> v_A² + v_B² + 2v_Av_B = v_A² + v_B²

=> 2v_Av_B = 0

which is possible only if v_B or v_A is = 0. The first would require that B is affected by a force without any change in velocity (impossible) so the latter is true.

Inelastic collision