Mechanics 1

In order to create a simple mathematical model we may have to make some assumptions.

Particle - has no size but does have a mass, this means that all forces will act in one place.

Rigid body - has a size but does not change shape when forces are applied to it.

Some things are ignored otherwise the model becomes too complicated, such as: air resistance, friction, resistance in a pulley, mass of a string and the elasticity of a string.

Keywords

• Smooth - no friction
• Rough - friction present
• Light - has no mass
• Inelastic - does not stretch
• Inextensible - does not stretch

Making assumptions will decrease the validity of the answers.

S displacement U initial velocity V final velocity A acceleration T time formula:

• v = u + at
• s = ut + 0.5at 2
• s = 0.5 (u +v)t
• v 2 = u 2 +2as

Displacement - distance from a specific origin, taking into account the direction of travel

Velocity - speed with a direction, ms -1

Acceleration - the rate at which the velocity changes, ms -2

Displacement-Time graphs: gradient is the velocity

Velocity-Time graphs: gradient is the acceleration, area under the graph is the displacement

Speed = Distance/Time            Velocity = Displacement/Time

R position at time t U initial velocity V velocity at time t A acceleration R0 initial position formula:

• r = ut + 0.5at2 + r0
• v = u + at
• r = 0.5(u + v)t + r0

r = xi + yj is the same as [x/y]

i is horizontal, j is vertical

remember trig and pythagoras, cosine rule a2 = b2 + c2 - 2bc cos A, sine rule sinA/a = sinB/b = sinC/c

p = ai + bj    q = ci + dj                                p = [a/b]              q = [c/d]

p + q = (a + c)i + (b + d)j                             p + q = [a + c/ b + d]

p - q = (a - c)i + (b - d)j                               p - q = [a - c/ b - d]

kp = kai + kbj                                              kp = [ka/kb]

W = mg N      g = 9.8ms-2        F = µR

 R - the normal reaction force, it acts perpendicular to the ground, it stops objects sinking into the ground.

Tension - a force exerted by a taut string, it can only pull and not push.

Resultant forces - the sum of two or more forces, triangles! trig, cosine rule and sine rule

Equilibrium - the forces must be equal and opposite, the sum of the resultant forces is zero, this happens when a body is at rest or moving with a constant velocity 

• F = ma

F = resultant force      m = mass       a = acceleration

• For every action there is an equal but opposite reaction

• When a body is at rest or moves with a constant velocity the forces must be in equilibrium

An equation of motion should be formed for each particle, using Newton's second law. The two equations can be solved by using standard techniques for simultaeneous equations.

Don't forget about tension! the tension has the same magnitude throughout the string

F is the resultant force:

      F   = ma

mg - T = ma

Assumptions :

• the object is a particle.
• gravity is the only force that acts on it.
• the object is set in motion so that it has a specific initial velocity

horizontal displacement x = V cos 0t      vertical displacement y = V sin 0t - 0.5gt2

V = initial speed      

horizontal velocity vx = V cos 0    vertical velocity vy = V sin 0 - gt

Range -horizontal distance travelled by projectile

At its max height the vertical velocity is zero

P = mv   P - momentum

Conservation of momentum in 1D:

P before = P after

Conservation of momentum in 2D:

 maua + mbub = mava + mbvb

u = initial velocity      v = final velocity

Coalesce - come together to form one unit