Tuesday 8 June 2010

Mechanics notes I

Caution:
1) In order to avoid the usage of equation editor, reader should be careful about whether the physical quantities are a vector or scalar.
2) The concept of calculus is used.
3) Graph is not given in the sense that it is driven from a set of datum. However the concept will be discussed. Readers may try to draw the graph themselves.
4)Sorry that I have no time to make the power/subscribed words/bolded set/...

Time (t): SI unit second (s), reaction time of human = 0.2s
Length: SI unit metre (m).
Scalar: A physical quantity describes the magnitude only. Mathematically, x∈R.
Vector: A physical quantity that gives size and direction. Mathematically x∈R2, Usually we will add an arrow on the physical quantities to mark that it is a vector. Another representation will be V=(a,b), a,b∈R. Note that |v|=(a2+b2)0.5, and it’s direction will be tan-1(a/b). Also (a,b)+(c,d)=(a+c,b+d).
Distance (s) (scalar) is defined as the total length of the path, with SI unit m.
Displacement (s) (vector) measures the change in position.
Displacement on a 1D case: Definition of positive direction, the displacement will be a scalar, where the +ve/-ve sign denotes the direction.
Displacement on a 2D case: Define every straight line travelled as s1=(x1,y1) to sn(xn,yn), then distance travelled = Σ(|sn|) while displacement = |(Σsn)|
Average speed (v) = (distance) s/t which is a scalar with SI unit m s-1.
Average velocity (v) = (displacement) s/t which is a vector with SI unit m s-1.
Considering the velocity on 1D/2D is similar to the displacement.
Instantaneous speed: average speed in a very short time. Mathematically it is Δs (distance) /t
Instantaneous velocity: average velocity in a very short time. v=ds (displacement) /dt
Uniform motion: when v(t) is a constant we call the motion of the object as uniform.
We define final velocity as v while initial velocity is u. Then the average acceleration over a time period is defined as a=(v-u)/t. The instantaneous acceleration will be a=dv/dt.
Uniformly acceleration: if a(t) is a constant, then we say it is uniformly accelerated. By F=ma if it is uniformly accelerated then the force is constantly given.
Displacement-time graph (s-t graph): y=s(t). y’=v(t).
Velocity-time graph (v-t graph): y=v(t), y’=a(t), ∫y dt=s(t).
Acceleration-time graph (a-t graph): y=a(t), ∫y dt=v(t).
Equations on uniform motion: (Under vertical motion, a is fixed as a=g=10ms-1.
1) s=(u+v)t/2, which can be proven by ∫y dt=s(t).
2) v=u+at, which can be proven by ∫y dt=v(t)
3) s=ut+at2/2 since s=(u+v)t/2=(2u+at)t/2=ut+at2/2.
4) v2=u2+2as since v-u=at, so (v+u)(v-u)=at(v+u)=2as.
Note: In vertical motion, neglecting air resistance, the s-t graph is symmetric and is a parabola.
Newton’s First Law of Motion: An object will remain at rest or constant velocity as long as the net force acting on it is zero.
Note that net force = ΣF which is the sum of all force acting on it.
Net force can be zero if the several force is balanced, or no force acting on it.
Inertia: the tendency of an object to remain at rest or in uniform motion, measured in mass and the unit is kg. It is harder to move an object with larger inertia.
Force (vector) (F): can be measured by spring balance/force sensors, with SI unit Newton (N).
Force can be divided into contact force or non-contact force. Some common force will be the weight (W=mg), normal reaction (R), friction (Ff) and tension (T).
Resultant force = ΣF which is the sum of given forces.
Calculation on force refers to the calculations of vectors.
Notes on free-body diagram:
1) Weight: vertically downwards from the center of mass
2) Normal reaction: pointing up from the contact surface, perpendicular to the surface. If the surface is an inclined plane, then the starting point should be the projection of center of mass vertically downwards to the contact surface.
3) Friction: On the contact surface.
Newton’s Second Law of Motion: a is proportional to ΣF (same direction) and inversely proportional to m. Considering the SI unit, then F=ma.
Application:
1) Weight: W=mg
2) Apparent weight during acceleration: The apparent weight will be R (not W); by ΣF=W-R=ma, we have R=W-ma=m(g-a) which implies that when the acceleration has the same direction with the gravity, our apparent weight will be reduced.
3) Friction (Max Ff=μR, where μ is the friction constant) appears when two surfaces slide or tends to slide over each other. When ΣF ≧ Max Ff it will start to move. Note that the max. (static) friction is a bit larger than the kinetic friction.
4) Fluid (liquid/gas) resistance increases as the difference between velocity and terminal velocity increase. The velocity of free-fall object will tends to the terminal velocity.
Newton’s Third Law of Motion: For every action there exist an equal but opposite reaction.
The action-reaction pair will be the force acting on A by B and the force acting on B by A.