Difference between revisions of "Forest UCM MiNF"
Line 142: | Line 142: | ||
The three forces influencing this drop from the previous sections are | The three forces influencing this drop from the previous sections are | ||
− | :<math>\vec F_g = | + | :<math>\vec F_g =M_o \vec g \;\;\;\; \vec{F}_{\mbox{tide}} \;\;\;\;\; \vec F_N</math> |
;Archimedes Principle | ;Archimedes Principle | ||
Line 149: | Line 149: | ||
All of the above forces act normal to the surface of the water. | All of the above forces act normal to the surface of the water. | ||
− | :<math>\vec F_g = | + | :<math>\vec F_g = M_o \vec g \;\;\;\; \vec{F}_{\mbox{tide}} </math> |
are the result of a gravitational force which is conservative. | are the result of a gravitational force which is conservative. | ||
Line 155: | Line 155: | ||
A potential may be defined for the two forces above such that | A potential may be defined for the two forces above such that | ||
− | :<math>\vec F_g = | + | :<math>\vec F_g = M_o \vec g = -\nabla U_{eg} \;\;\;\; \vec{F}_{\mbox{tide}} = - \nabla U_{\mbox{tide}} </math> |
Line 176: | Line 176: | ||
Since | Since | ||
− | :<math>\vec F_g = | + | :<math>\vec F_g = M_o \vec g = -\nabla U_{eg} \;\;\;\; \vec{F}_{\mbox{tide}} = - \nabla U_{\mbox{tide}} </math> |
are forces that are exerted such that the are always perpendicular to the surface of the ocean ( they are normal to the surface) then the sum of the two potential's, <math>U_{eg} + U_{\mbox{tide}}</math>, is constant on the surface | are forces that are exerted such that the are always perpendicular to the surface of the ocean ( they are normal to the surface) then the sum of the two potential's, <math>U_{eg} + U_{\mbox{tide}}</math>, is constant on the surface | ||
Line 189: | Line 189: | ||
− | :<math> | + | :<math>M_o gh=U_{eg}(HT)-U_{eg}(LT)</math> |
but | but | ||
Line 205: | Line 205: | ||
− | :<math> | + | :<math>M_ogh= -G M_m M_o \left [ \left ( \frac{1}{\sqrt{d_0^2+r^2}} \right )- \left ( \frac{1}{d_0-R_e} + \frac{-R_e}{d_0^2} \right ) \right ]</math> |
::<math>= -G M_m M_o \left [ \frac{1}{d_0} \left ( 1 - \frac{1}{2} \left ( \frac{R_e}{d_0} \right )^2 \right ) - \left ( \frac{1}{d_0-R_e} + \frac{-R_e}{d_0^2} \right ) \right ]</math> | ::<math>= -G M_m M_o \left [ \frac{1}{d_0} \left ( 1 - \frac{1}{2} \left ( \frac{R_e}{d_0} \right )^2 \right ) - \left ( \frac{1}{d_0-R_e} + \frac{-R_e}{d_0^2} \right ) \right ]</math> | ||
::<math>= -G M_m M_o \left [ \frac{1}{d_0} \left ( 1 - \frac{1}{2} \left ( \frac{R_e}{d_0} \right )^2 \right ) - \frac{1}{d_0}\left ( \frac{1}{1-\frac{R_e}{d_0}} + \frac{-R_e}{d_0^2} \right ) \right ]</math> | ::<math>= -G M_m M_o \left [ \frac{1}{d_0} \left ( 1 - \frac{1}{2} \left ( \frac{R_e}{d_0} \right )^2 \right ) - \frac{1}{d_0}\left ( \frac{1}{1-\frac{R_e}{d_0}} + \frac{-R_e}{d_0^2} \right ) \right ]</math> | ||
Line 213: | Line 213: | ||
− | :<math> | + | :<math>M_o gh= -G M_m M_o \left [ \frac{1}{d_0} \left ( 1 - \frac{1}{2} \left ( \frac{R_e}{d_0} \right )^2 \right ) - \frac{1}{d_0}\left (1 + \frac{R_e}{d_0} + \left( \frac{R_e}{d_0} \right)^2 - \frac{R_e}{d_0^2} \right ) \right ]</math> |
::<math> -G M_m M_o \left [ \frac{1}{d_0} \left ( 1 - \frac{1}{2} \left ( \frac{R_e}{d_0} \right )^2 \right ) - \frac{1}{d_0}\left (1 + \left( \frac{R_e}{d_0} \right)^2 \right ) \right ]</math> | ::<math> -G M_m M_o \left [ \frac{1}{d_0} \left ( 1 - \frac{1}{2} \left ( \frac{R_e}{d_0} \right )^2 \right ) - \frac{1}{d_0}\left (1 + \left( \frac{R_e}{d_0} \right)^2 \right ) \right ]</math> | ||
::<math> -G M_m M_o \left [ \frac{1}{d_0} \left ( - \frac{3}{2} \left ( \frac{R_e}{d_0} \right )^2 \right ) \right ]</math> | ::<math> -G M_m M_o \left [ \frac{1}{d_0} \left ( - \frac{3}{2} \left ( \frac{R_e}{d_0} \right )^2 \right ) \right ]</math> |
Revision as of 19:23, 7 November 2014
Mechanics in Noninertial Reference Frames
Linearly accelerating reference frames
Let
represent an inertial reference frame and \mathcal S represent an noninertial reference frame with acceleration relative to .Ball thrown straight up
Consider the motion of a ball thrown straight up as viewed from
.
Using a Galilean transformation (not a relativistic Lorentz transformation)
At some instant in time the velocities add like
where
- = velocity of moving frame with respect to at some instant in time
taking derivative with respect to time
where
- inertial force
- in your noninertial frame, the ball appears to have a force causing it to accelerate in the direction.
The inertial force may also be referred to as a fictional force
an example is the "fictional" centrifugal force for rotational acceleration.
The observer in a noninertial reference frame will feel these frictional forces as if they are real but they are really a consequence of your accelerating reference frame
example
- A force pushes you back into your seat when your Jet airplane takes off
- you slam on the brakes and hit your head on the car's dashboard
Pedulum in an accelerating car
Consider a pendulum mounted inside a car that is accelerating to the right with a constant acceleration
.What is the pendulums equilibrium angle
In frame
In frame
If the pendulum is at rest and not oscillating then
is the vector sum of and which are orthogonal to each other in this problem thus
The pendulum oscillation frequency as seen in the accelerating car is
- Using lagrangian mechanics in the inertial frame
The tides
The gravitational force between the moon and the earth accelerates the earth and the ocean towards the moon.
The moon of mass
pulls on the earth of mass such thatwhere
is the earth-moon distance of separation.- Earth's acceleration towards the moon that makes the earth a non-inertial reference frame
The moon of mass pulls on a test mass of water on the surface of the earth's ocean such that
As seen in the Earth non-inertial reference frame
where
- a net non-graviational force hold the mass M_o in top of the ocean (Bouyant force)
Let's consider two cases, one where M_o is directly between the moon and the earth and the other when the mass is directly on the opposite side of the earth from the moon.
- Case 1
- The mass is directly between the moon and the earth
- In this case and making pull towards the moon
- Case 2
- The mass is directly on the other side of the earth with respect to the moon
- In this case BUT making pull away from the moon
Magnitude of the Tides
Consider a drop of water of mass
on the surface of the ocean.The three forces influencing this drop from the previous sections are
- Archimedes Principle
- An object in a fluid is buoyed up with a force equal to the weight of the water displaced by the object.
All of the above forces act normal to the surface of the water.
are the result of a gravitational force which is conservative.
A potential may be defined for the two forces above such that
- : is NOT always pointed along the x-axis towards the moon
- Since is at fixed distance
- Since is parallel to x
Since
are forces that are exerted such that the are always perpendicular to the surface of the ocean ( they are normal to the surface) then the sum of the two potential's,
, is constant on the surface- The ocean's surface is an equipotential surface ( all points on the surface of the ocean are have the same gravitational potential energy
If you consider two points on the surface of the earth where one point is high tide (HT) and the other point is low tide (LT) then
The change in the gravitational potential energy between the earth and the ocean for high tide and low tide is
but
At HT
At LT
- : geometric series when