Mechanics notes II

*This set of notes act as a remainder rather than the actual application on different problems. Readers should try themselves in different problems as well.
I will complete the DSE syb. of mechanics except gravitation this year. This include:
I: Position and moment, Newton's Law of Motion
II: Fotce in a plane and moment, work, energy and pwoer
III: Momentum, projectile and circular (this one is much more easier than AL)

Methods to add vectors up:
Tip-to-tail method: Consider two forces F1 and F2. Firstly draw the vector F1 starting from the origin, and draw F2 from the tail of F1. The tail of F2 will be the resultant force.
Parallelogram of forces: Draw the two forces at the origin, project them to another’s tail. The intersection point will be the tail of resultant force.
Consider F is a force with magnitude F and angle between the force and the x-axis is θ. Then it can be resolved into Fx=Fcosθ and Fy=Fsinθ. Mathematically F==F(cosθI+sinθj).
When we apply Newton’s Law of Motion on a plane, we always resolve the force along the plane and perpendicular to the plane. Then each of the forces will never be offset by each other. Also note that when the object is at rest or in uniform motion then the net force is zero.
Moment or a torque at a point by a force is defined as Fd, which is the force times he perpendicular distance from the point. Note that if the fixed point, point giving force and the direction of force is parallel then its moment will be zero since it can’t rotate the object. Alternatively if the force makes an angle θ with the object then the perpendicular distance will be dsinθ. Comparing gives the moment = Fdsinθ. Moments can have 2 direction: clockwise or anti-clockwise.
The principle of moments states that the object is balanced if the total clockwise moments is equal to that of anti-clockwise moments.
When two forces with equal magnitudes acting in opposite directions on an object, they will form a couple, which produce a moment of force(one force, not the sum of the two forces) times perpendicular distance between the two forces, no matter which point is fixed to calculate the moment.
Equilibrium is defined as net force acting on it is zero and net moment at any point of it is zero.
Centre of gravity (c.g.) is the point through which the weight of the whole object acts. If any object is hanged, then it will move until the c.g. is vertically below the fixed point. Otherwise a couple is produced to rotate that object to that stable case.
There will be three types of equilibrium: Stable equilibrium implies that the c.g. have to move upward in order to move it. Unstable equilibrium implies that the c.g. will fall as to make the whole object moves. Neutral equilibrium implies that the vertical position of c.g. won’t change even the position of the whole object changes.
Doing work means the process of energy transfer.
Work done by a force = component of force in the direction of displacement times displacements.
If the forces makes an angle θ with the displacement, then the component will be Fcosθ. Mathematically W=Fscosθ. It is a scalar with unit Joule (J).
Note that W=0 if (1) s=0 or (2) cosθ=0, then the force is perpendicular to the displacement, which the component of force with the same direction with the displacement is zero.
Work done by a force is also the energy transferred by the force. A positive work done means that the object gains mechanical energy, and a negative work done means that it losses mechanical energy (but internal energy can be gained).
Now consider the object moving in a straight line (therefore consider K.E. only):
W=Fs=(ma)(v2-u2/2a)=mv2/2-mu2/2=ΔK.E., therefore we can identify that K.E. of an object is mv2/2 where v is its velocity.
Potential energy can be divided into elastic potential energy (e.p.e.) and gravitational potential energy (g.p.e.). e.p.e. is stored in an elastic object when it is stretched, compressed or bent. When it is released the energy will be released too.
For g.p.e. we have ΔP.E.=mgΔh, we can’t determine the g.p.e. of an object since the “height” is a relative concept, but Δh can be accurately measured.
The conservation of mechanical energy states that initial K.E.+ initial P.E.=final K.E.+ final P.E.
That is, the mechanical energy of an object is conserved, provided that no work done is on the object. This implies that:
1) If mg is the only force acting on it, then ΔKE+ΔPE=0.
2) If the resultant force is mg or component of mg, then ΔKE+ΔPE=0. This mainly happens when the object is on a inclined plane, where part of the mg offsets with R.
3) In other case, ΔKE+ΔPE = work done by the forces other than the (component) of gravitational forces.
In a simple pendulum, v is maximized when it is in the lowest position where we can assume it’s PE is relatively zero.
The law of conservation of energy states that energy can’t be destroyed or created. But it can transfer from one to another.
Power is the rate of energy transfer or work is done. Mathematically P=E/t=W/t. It is a scalar with unit Watt (W). Also, P=W/t=Fs/t=F(s/t)=Fv. Therefore a power P it required to apply a force to an object such that it is moving in constant velocity v. (Note that if it is moving in constant velocity, there’ll be another force opposite to this force.)