2012 Notes on Physics and Calculus

Wednesday, September 5, 2012

Significant Physics Constants

Sources

$\epsilon_0 \approx 8.854187817620... 10^{-12} $

Source = Wikipedia
$\mbox{Used in Coulomb's Law defining the electrostatic force}$

$F_c = \left(\dfrac{1}{4 \pi \epsilon_0}\right) \left(\dfrac{q_1 q_2}{r^2}\right)$

$\epsilon_0 \mbox{is defined by:}$

$\epsilon_0 = \dfrac{1}{\mu_0 c^2} $

$\mu_0 = 4 \pi 10^{-7} \approx 1.2566370614... 10^{-6}$

Source = Wikipedia
In electromagnetism, permeability is the measure of the ability of a material to support the formation of a magnetic field within itself.

$\mbox{Gravitational constant} \; G = 6.67384 10^{-11} $

Source = Wikipedia
According to the law of universal gravitation, the attractive force (F) between two bodies is proportional to the product of their masses (m₁ and m₂), and inversely proportional to the square of the distance, r, (inverse-square law) between them:

$F_g = G \dfrac{m_1 m_2}{r^2} $

Tuesday, August 7, 2012

Final 17 - Springs and Tables

Given:

$\mbox{A 1 kg mass is connected to a spring with a constant of 350 newtons/meter.} $
$\mbox{The spring is compressed 25 centimeters and let go from 2 meters.}$

Question:

$\mbox{How far away will the mass impact the floor?} $

Rationale:

$PE_{spring} = KE_{mass} $

$.5 K x^2 = .5 m V_x^2 $
$V_x^2 = \dfrac{(.5 * K * x^2 )}{(.5 * m)}$

$t_y^2 = \dfrac{2 * \Delta{y}}{g} $
$t_y = t_x $
$\Delta{x} = V_x * t_x $

$\Delta{x} = \sqrt{\dfrac{(.5 * K * x^2 )}{(.5 * m)}} * \sqrt{\dfrac{2 * \Delta{y}}{g}} $

Calculate:

$\Delta{x} = \sqrt{\dfrac{(.5 * 350 * .25^2 )}{(.5 * 1)}} * \sqrt{\dfrac{2 * 2}{10}} $

$\Delta_{x} = \sqrt{10.9375 * .5} * \sqrt{.4} = 4.6770717 * .632455 = 2.9580374$
$\Delta_{x} \approx 2.96 \; \mbox{meters}$

References:

$ $

Final 16 - Bikes and Buses

Final 15 - Gravity and Electricy

Given:

$\mbox{A mass of 8 kg and a charge of 0.1 coulombs}$

$\mbox{Is dropped from a height of 1.2 meters}$
$\mbox{In the continuing presence of an electric field of 100 newtons/coulomb.} $

Question:

$\mbox{What distance d does the mass hit the floor?} $

Rationale:

$\Delta{y} = 0.5 g t^2 $

$ t^2 = \dfrac{2 \Delta{y} }{g} $

$F_{electric} = F_x = m a_x $

$E q = m a_x $
$a_x = \dfrac{(E q)}{m} $

$t_y^2 = t_x^2 $
$\Delta{x} = .5 (\dfrac{(E q)}{m}) \dfrac{2 \Delta{y} }{g} $

Calculate:

$\Delta{x} = .5 (\dfrac{(100 * .1)}{8}) \dfrac{2 * 1.2 }{10} = 0.15 \; \mbox{meters}$

References:

$ $

Final 14 - Pushing A Car

Given:

$\mbox{A car is being pushed by three equal force with one on each side at 30 degrees and one in the rear.} $
$\mbox{The 1000 kg car is being accelerated at 0.5 m/s/s by the efforts of these three.} $

Question:

$\mbox{What force are each exerting on the car?} $

Rationale:

$F = F_1 = F_2 = F_3 $
$F_y = F \cos\alpha \; \mbox{(and there are two of these forces)} $

$F_{total} = F + (2 F \cos\alpha) $

$F_{car} = m a = F + (2 F \cos\alpha) $
$ F (1+(2 \sin\alpha) = m a $
$ F = \dfrac{( m a)}{(1+(2 \cos\alpha)} $

Calculate:

$F = \dfrac{(100 * .5)}{(1 + (2 * .866025)} = \dfrac{500}{2.73205} = 183.012756 \approx 183$

References:

$ $

Final 13 - Double Inclined Plane

Given:

$\mbox{Two masses connected by a rope and pully} $

$\mbox{mass 1 is on a plane inclined at an angle alpha} $
$\mbox{mass 2 is on a plane inclined at an angle beta} $

$a = g \frac{M_{unknown}\sin{\beta} {(operator)} M_{unknown}\sin{\alpha}}{M_1 ({operator}) M_2} $

Question:

$What are subscripts and operators? $

Rationale:

$M_1 \; \mbox{is associated with angle alpha} $
$M_2 \; \mbox{is associated with angle beta} $

$a = g \frac{M_2 \sin{\beta} {(operator)} M_1 \sin{\alpha}}{M_1 ({operator}) M_2} $

$\mbox{The denominator operator cannot be multiply or divide}$

$\mbox{Since if either mass is zero, then the equation would be indeterminate} $
$\mbox{So the operator must be + or - }$

$\mbox{But it cannot be minus, since it would indeterminate is the masses are equal}$
$\mbox{Therefore the denominator operator is plus.}$

$\mbox{If the masses are equal and the angles are equal, }$

$\mbox{Then the numerator must be zero}$
$\mbox{Thus the numerator operator must be minus.}$

Calculate:

$ $

References:

$ $

Final 12 - Water Clock

Given:

$\mbox{In one time interval, a ball rolls down the incline 0.8 meters.} $

Question:

$\mbox{What interval does the ball travel in the second and third time intervals?} $

Rationale & Calculate:

$\Delta{x} \varpropto t^2 $

$\Delta{x-1} = 1 * K $
$\Delta{x-2} = 4 * K$
$\Delta{x-3} = 9 * K $

$ K = 0.8 \; \mbox{since at t-1, x-1 is 0.8} $

$\Delta{x-2} = 3.2 $
$\Delta{x-3} = 7.2 $

$ Interval_{2-1} = 3.2 - 0.8 = 2.4 $
$ Interval_{3-2} = 7.2 - 3.2 = 4.0 $

References:

Unit 2A-8 of PH-100

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