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.
- 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.
Kinematics in 1D
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
Kinematics in 2D
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
Newton's Laws of Motion
- 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
- 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