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2. Mechanics

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2.1 Motion

Distance and displacement

Distane Displacement
scalar vector
The length of the shortest straight line between two different point The distance moved from the initial position to the end point
|s| s

distance_and_velocity.jpeg

Speed and Velocity

Speed Velocity
Scalar Vector
How fast an object moves/rate of change of distance to time Speed with the direction/rate of change of displacement to time
|v| v
The derivative of displacement is velocity

Acceleration

  • vector
  • Describes how fast the speed of an object changes/The rate of change of velocity to time
  • The derivative of velocity is acceleration
Equations of motion for uniform acceleration
${v}=ut+\frac{1}{2}at^2$
${s}=ut+\frac{1}{2}at^2$
${v^2}=u^2+2as$
${s}=\frac{(u+v)t}{2}$
v=velocity, s=displacement, a=acceleration, t=time

Projectile motion

Projectile motion is The projectile motion is the uniform speed curve motion in which the object is launched from a certain initial angle, ignoring the air resistance, and the object is only affected by gravity (g) and initial velocity

projectile_motion.jpeg

  • The air resistance is negligible
  • The horizontal component of velocity is constant
  • The verticle component of velocity has acceleration downward of g(9.81ms^-2)
  • The motion's is zero when it at the heighest point
  • The motion is symmetric

| The verticle and horizontal component |

Horizontal component Verticle component
u~x~ u~y~
v~x~=u~x~ v~y~=u~y~ - gt
v~x~=ucosθ v~y~=usinθ - gt

component.jpeg

2.2 Forces

Different kinds of force

Forces and their direction {.tabset}

Weight

Weight
${W}=mg$
Where W is weight, m is mass and g is gravity constant(9.81ms^-2^ on Earth)
The weight is measurement of gravitational attraction between the mass m of a body and the mass of the planet on which the body is placed
weight.jpeg

Tension

Tension
When two forces are applied to the rope, it will tight up and have tension. PS. only the tight rope has tension
tension.jpeg

Forces in springs

Forces in springs
${T}=kx$
Where T is spring force, k is spring constant and x is the displacement between the equilibrium point and release point
When there's displacement of spring, the spring will store the force and try to back to the original value
spring_force.jpeg

Drag forces

Drag forces
Drag forces are forces that oppose the motion of a body through a fluid(gas or liquid)
drag_force.jpeg

Upthrust

upthrust
Any object placed in a fluid experiences an upward force which is upthrust
upthrust.jpeg

Frictional forces

Frctional forces
dynamic friction static friction
${fd}=μdR$ ${fs}=μsR$
Where f is frictional forces, μ is coefficient of dynamic friction and R is the normal reaction force between the surfaces Where f is frictional forces, μ is coefficient of static friction and R is the normal reaction force between the surfaces
The force of dynamic friction dose not depend on the speed of sliding The force of static friction depend on the speed of sliding
${μs}>μd$
frictional_forces.jpeg

Free-body diagrams & equilibrium

Free-body diagram shows the magnitude and direction of all the forces acting on a chosen body, and in this graph we treated body as point mass (which have no volume) free_body_diagram.jpeg Equilibrium refers to a points which net force(sum of all it's component is zero) like the situation: · object moving in constant velocity · object stay rest equilibrium.jpeg

Newton's law of motion

Newton's first law
When the net force on a body is zero, the body will move with constant velocity(which may be zero)
Newton's third law
If a body A exerts a force on body B, then body B exerts a force of the same magnitude but in the opposite direction of body A.
Newton's second law
The net force on a body of constant mass is proportional to that body's acceleration and is in the same direction as the acceleration
F=ma(very improtant formula)

2.3 Work, energy and power

Work done

The work done by a force is the product of the force in the direction of the displacement time and distance travel Follow the equation ${W}=Fscosθ$

Energy

Kenetic energy

$E_K=\frac{1}{2}mv^2$

$E_K$ is kenetic energy, m is mass, v is velocity of the object

Gravity potential energy

$E_P=mgh$

$E_P$ is gravity potiential energy, m is mass, h is height and g is gravity constant(9.81ms^-2^ on Earth)

Mechanical energy

$E_P=\frac{1}{2}x^2$

$E_P$ is elastic potiential energy, m is mass, x is displacement.

Power

$P=\frac{W}{t}$/$P=Fv$ Where P is power, W is work done and t is time, in the other formula, F is forces and v is velocity

Power 衡量的是规定时间内做工的能力

Efficiency

efficitency.webp

2.4 Motion

Newton's second law in terms of momentum
$p=mv$
$F_{net}=\frac{\Delta p}{\Delta t}$
p is momentum, m is mass and v is velocity
The change of momentum of an object is called impulse
Rearranging the formula describing Newton’s second law results in the following expression
F~net~${\Delta t}$=$m{\Delta v}$
left side is impluse and right side is change in momentum

Impluse and force-time graphs

Impluse is the area in the force-time graph force-time_graph.jpeg

conservation of momentum

When the net force on a system is zero the momentum does not change

Follow the equation: $m_1u_1+m_2u_2=m_1v_1+m_2v_2$

Where m is mass, u is initial velocity and v is final velocity