W3 Circular Motion and Energy

Calculator: only Casio FX82

• the only calculator permitted in University of Melbourne exams if the Casio FX82 (you have it)

Friction

• 

Drag

• a resistive force on an object moving in a fluid (gas or liquid)
• $\uparrow v \implies \uparrow \vec{D}$
• $\boxed{\vec{D}=\frac{1}{2}C\rho Av^2}$
  • $C$ = drag coefficient (often $C=\frac{1}{2}$)
  • $\rho$ = density of fluid (for air: $1.29kg/m^3$)
  • $v$ = velocity
  • $A$ = cross-sectional area of object/area that faces the wind
• $A$:
  • 
  • 

Terminal Velocity

• when $a$ (acceleration) = 0
• can be derived from drag formula
• $\boxed{v_{term}=\sqrt{\frac{2mg}{C\rho A}}}$
  • $C$ = drag coefficient (often $C=\frac{1}{2}$)
  • $\rho$ = density of fluid ($kg/m^3$);for air: $1.29kg/m^3$
  • $v$ = velocity ($m/s$)
  • $A$ = cross-sectional area of object/area that faces the wind ($m^2$)

Uniform Circular Motion

arc length

• $\boxed{l=r\theta}$
  • $l$ = arc length ($m$)
  • $\theta$ = angle in ($rad$)
  • $r$ = radius ($m$)
• 

velocity

• $\boxed{|\vec{v}|=v=\frac{\Delta x}{\Delta t}=\frac{2\pi r}{T}=\omega r}$
• speed of an obejct moving in circular motion is constant but velocity is always changing
  • because of change of direction but no change in magnitude
• Period and Frequency
  • $T$ = period ($s$); time taken for object to go around the circle once
  • $f$ = frequency ($Hz$ or $s^{-1}$) = $\frac{1}{T}$
• Angular
  • $\omega$ = angular velocity = $\frac{2\pi}{T}$ ($rad/s$)
  • $\alpha$ = angular acceleration = $\frac{\Delta \omega}{\Delta t}$

centripetal (net) acceleration

• constantly changing - always points to center of the circle
• 
• $\boxed{\vec{a_c}=a=\frac{v^2}{r}}$
  • $a_c$

centripetal net force

• $\boxed{\vec{F}_{net}=\frac{mv^2}{r} }$
  • $\vec{F}_{net}$ = centripetal net force; toward the center of the circle
  • $m$ = mass ($kg$)
  • $v$ = velocity ($m/s^2$)
  • $r$ = radius ($m$)

Torque

• $\boxed{\tau=rF\sin\phi}$
  • $\tau$ = torque ($N$)
  • $r$ = distance from axis of rotation to point where force is applied ($m$)
  • $F$ = Force applied ($N$)
  • $\phi$ = angle between foce and line drawn from axis to point where force is applied ($\degree$ or $rad$)
    • maximum when $90\degree$
  • ;

angular acceleration

• $\boxed{\alpha=\frac{\tau}{I}=\frac{\tau}{mr^2}}$
  • $\alpha$ = angular acceleration
  • $\tau$ = torque ($N$)
  • $I$ = moment of inertia = $mr^2$

Static Equilibrium

• net force is zero
  • $F_{net}=0$
• net torque about any axis is zero
  • $\tau_{net}=0$

Newton’s law of gravitation

• $\boxed{F_{\textmd{1 on 2}} = F_{\textmd{2 on 1}}=\frac{Gm_1m_2}{r^2}}$
  • 
  • $G=6.67\times10^{-11}Nm^2kg^{-2}$

Orbits

• $\boxed{v=\sqrt{\frac{Gm_2}{r}}}$

Weightlessness in orbit

• 

Conservation laws in physics

• conservation of mass
• conservation of momentum
• ⚠️ conservation of energy
• conservation of angular momentum

Collissions

• a short-duration interaction between two objects

Impulse and Linear Momentum

• Impulse
  • area under force-time graph
  • $\boxed{J=\int_{t_{i}}^{t_{f}}Fdt=F_{av}\Delta t}$
• Linear momentum
• $\boxed{\vec{p}=m\vec{v}}$
  • $\vec{p}$ = momentum ($kg m/s$)
  • $m$ = mass ($kg$)
  • $\vec{v}$ = velocity ($m/s$)
• Impulse-momentum theorem
  • $\boxed{\vec{J}=\vec{F}_{av}\Delta t=m\Delta\vec{c}=\Delta\vec{p}}$

Conservation of Momentum

• $\boxed{\vec{F}_{\textmd{net ext}}=\frac{d\vec{P}}{dt}}$
• isolated: if $\vec{F}_{\textmd{net ext}}=0$, total momentum is conserved (ie. has a constant value)
• $F_{\textmd{net ext}_{ave}}=\frac{\Delta P}{\Delta t}$
• $\Delta P = F_{net}\Delta t$