משתמש:Coredumb~hewikibooks/ארגז חול

אחר כך.

דף זה מהווה סקירה של נוסחאות חשובות במכניקה הקלאסית.

מונחים

a = acceleration (m/s²)

g = gravitational constant (m/s²)

F = force (N = kg m/s²)

E_k = kinetic energy (J = kg m²/s²)

E_p = potential energy (J = kg m²/s²)

m = mass (kg)

p = momentum (kg m/s)

s = position (m)

R = radius (m)

t = time (s)

v = velocity (m/s)

v₀ = velocity at time t=0

W = work (J = kg m²/s²)

τ = torque (J = N m) (torque is the rotational form of force)

s(t) = position at time t

s₀ = position at time t=0

r_unit = unit vector pointing from the origin in polar coordinates

θ_unit = unit vector pointing in the direction of increasing values of theta in polor coordinates

Note: All quantities in bold represent vectors.

נוסחאות בסיסיות

מרכז מסה

In the discrete case:

$\mathbf {s} _{\hbox{CM}}={1 \over m_{\hbox{total}}}\sum _{i=0}^{n}m_{i}\mathbf {s} _{i}$

where $n$ is the number of mass particles.

Or in the continuous case:

$\mathbf {s} _{\hbox{CM}}={1 \over m_{\hbox{total}}}\int \rho (\mathbf {s} )dV$

where ρ(s) is the scalar mass density as a function of the position vecto

מהירות

$\mathbf {v} _{\mbox{average}}={\Delta \mathbf {s} \over \Delta t}$

$\mathbf {v} ={d\mathbf {s} \over dt}$

תאוצה

$\mathbf {a} _{\mbox{average}}={\frac {\Delta \mathbf {v} }{\Delta t}}$

$\mathbf {a} ={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}\mathbf {s} }{dt^{2}}}$

• Centripetal Acceleration

$|\mathbf {a} _{c}|=\omega ^{2}R=v^{2}/R$

(R = radius of the circle, ω = v/R angular velocity)

תנע

$\mathbf {p} =m\mathbf {v}$

כוחות

$\sum \mathbf {F} ={\frac {d\mathbf {p} }{dt}}={\frac {d(m\mathbf {v} )}{dt}}$

$\sum \mathbf {F} =m\mathbf {a} \quad \$   (Constant Mass)

מתקף

$\mathbf {J} =\Delta \mathbf {p} =\int \mathbf {F} dt$

$\mathbf {J} =\mathbf {F} \Delta t\quad \$

  if F is constant

מומנט ההתמד

For a single axis of rotation: The moment of inertia for an object is the sum of the products of the mass element and the square of their distances from the axis of rotation:

$I=\sum r_{i}^{2}m_{i}=\int _{M}r^{2}\mathrm {d} m=\iiint _{V}r^{2}\rho (x,y,z)\mathrm {d} V$

תנע זוויתי

$|L|=mvr\quad \$   if v is perpendicular to r

Vector form:

$\mathbf {L} =\mathbf {r} \times \mathbf {p} =\mathbf {I} \,\omega$

(Note: I can be treated like a vector if it is diagonalized first, but it is actually a 3×3 matrix - a tensor of rank-2)

r is the radius vector

מומנט

$\sum {\boldsymbol {\tau }}={\frac {d\mathbf {L} }{dt}}$

$\sum {\boldsymbol {\tau }}=\mathbf {r} \times \mathbf {F} \quad$

if |r| and the sine of the angle between r and p remains constant.

$\sum {\boldsymbol {\tau }}=\mathbf {I} {\boldsymbol {\alpha }}$

This one is very limited, more added later. α = dω/dt

אנרגיה

m is here constant.

$\Delta E_{k}=\int \mathbf {F} _{\mbox{net}}\cdot d\mathbf {s} =\int \mathbf {v} \cdot d\mathbf {p} ={\begin{matrix}{\frac {1}{2}}\end{matrix}}mv^{2}-{\begin{matrix}{\frac {1}{2}}\end{matrix}}m{v_{0}}^{2}\quad \$

$\Delta E_{p}=mgh\quad \ \,\!$ in field of gravity

Central Force Motion

${\frac {d^{2}}{d\theta ^{2}}}\left({\frac {1}{\mathbf {r} }}\right)+{\frac {1}{\mathbf {r} }}=-{\frac {\mu \mathbf {r} ^{2}}{\mathbf {l} ^{2}}}\mathbf {F} (\mathbf {r} )$

כוח הכבידה

$\mathbf {F(r)} =-{\frac {\mathbf {Gm_{1}} \mathbf {m_{2}} }{\mathbf {r^{2}} }}$

G is the gravitational constant, one of the physical constants

נוסחאות הנובעות מנוסחאות הבסיס

Position of an accelerating body

$\mathbf {s} (t)={\begin{matrix}{\frac {1}{2}}\end{matrix}}\mathbf {a} t^{2}+\mathbf {v} _{0}t+\mathbf {s} _{0}\quad \$   if a is constant.

Equation for velocity

$v^{2}=v_{0}^{2}+2\mathbf {a} \cdot \Delta \mathbf {s}$