Karteikarten
Curved motion
Karten in diesem Stapel (42)
- 1displacement vector\[ \vec{r} = x \hat{i} + y \hat{j} + z \hat{k}\]
- 2Average velocity\[ \vec{\bar{v}} = \frac{\Delta \vec{r}}{\Delta t}\]
- 3Instantaneous velocity\[ \vec{v} = \frac{d \vec{r}}{dt}\]
In terms of the arc length \(s\)
\[ \vec{v} = \frac{ds}{dt} {\vec{u}}_T\]
where \( {\vec{u}}_T\) is a unitary vector tangent to the curve - 4velocity in curved motionIn gral:
1) \( \vec{v}\) changes in magnitud
2) and also in direction, since \(\vec{v}\) is
tangent to the curve - 5average and instant acceleration\[ \vec{\bar{a}} = \frac{\Delta {\vec v}}{\Delta t}\]
\( \vec{\bar{a}} \) is paralell to \( \Delta \vec{v}\)
\[ \vec{a} = \frac{d \vec{v}}{dt}\] - 6curved motion in constant accelerationFor \( \vec{a}\) constant the equations of motion are:
\[ \vec{v}= \vec{v_0} + \vec{a}(t-t_0)\]
\[\vec{r}=\vec{r}_0 +\vec{v}_0(t-t_0)+\frac{1}{2}\vec{a}(t-t_0)^2\] - 7curved motion with constant acceleration is in a planeFor \( \vec{r} = \vec{a}s^2 + \vec{b}s + \vec{c}\) with a,b,c constant vectors, the trajectory is parabolic on a plane given by
\( \vec {a} \) and \( \vec{b} \) that goes through a point with vector position \( \vec{c}\)
Then, for \( \vec{a}=const\), the motion is in a plane - 8Projectile motiona special case of curved motion with constant acceleration where
\[ a_x = 0,\,\, a_y = -g\] - 9Projectile motion:1) On \( x\)-axis the motion is rectilineo uniforme \( v_x = cte\)
2) On \( y\)-axis the motion is free-fall
with \( a_y = const.\) mov. uniformemente acelerado - 10ideal motion for the projectile1) no air resistance
2) no rotations of the object
3) \( a = -g\) is constant and downwards
4) no air velocity
5) R is enough small that one can reject the curvature of the Earth.
With this assumptions the path projectile is always an parabola - 11Equations of motion in x for the projectileon \(x\)-axis:
\[ x = x_0 + v_0 t \]
\[v_x = v_{0x}=const\] - 12Equations of motion in y for the projectile\[ y = y_0 + v_{0y}t + \frac{1}{2}a t^2\]
\[ y = y_0 + {\bar{V}}_y t \]
\[ v_y = v_{0y} + at\]
\[ \bar{V}_y= \frac{1}{2}(v_y + v_{0y})\]
\[ v_y^2 = v^2_{y0} + 2a(y - y_0)\] - 13velocity in y for the projectile1) the \( v_{0y}\) is upward and \( v_{Ry} = v_{y0}\) but it is downward
2) at the highest elevation \(v_y = 0\)
3) \( a_y \neq 0\) during the trajectory. - 14Maximum Heigth of a Projectileis given when \( v_{y} = 0\)
at a time
\[ t_H = \frac{v_{0y}}{g} \]
and is localized in the point \( (\frac{R}{2}, H)\) with \( R\) the horizontal range - 15the height H isis \[ H = y_0 + \frac{v_{y0}^2}{2g} \]
with \( v_{y0} = v_0 sen \theta_0\) - 16The horizontal range R\[ A = \frac{v_{0}^2}{g} sen(2 \theta_0)\]
maximum range is given for \( \theta_0 = 45\) degrees - 17Total time to reach the range\[ t = 2 \frac{v_{0y}}{g}\]
- 18For real projectileFor a projectile of large A (alcance), the trajectory is not parabolic but an arc of elipse.
There, the gravitational acceleration changes in direction and in magnitude with the altitude. - 19Tangential and normal acceleration in a curved trajectory\( \vec{a}_T\) : change in magnitud of the velocity
\( \vec{a}_N\):change in the direction of the velocity - 20Acceleration in curved trajectory\[ \vec{a} = \frac{d \vec{v}}{dt}\]
\[ = \vec{u}_T \frac{dv}{dt} + \frac{d \vec{u}_T}{dt}v \] - 21derivative of tangential unitary vector\[ \frac{d \vec{u}_T}{dt} = \vec{u}_N \frac{v}{\rho}\]
where \(\rho\) is the radius of curvature. - 22tangential and normal acceleration\[ \vec{a} = \vec{u}_T \frac{dv}{dt} + \vec{u}_N \frac{v^2}{\rho} \]
\[\vec{a}_N = \frac{v^2}{\rho}\]
\[ \vec{a}_T = \frac{dv}{dt}\] - 23mov. curvilineo uniforme\( |\vec{v}| = constant \) then \( a_T = 0\)
- 24mov, rectilíneothe direction of \(v\) does not change and
\( a_N = 0\) - 25Another case of curved motioncircular motion: when the trajectory is a circle.
- 26the velocity in circular motion\[ v = \frac{ds}{dt} = R \frac{d\theta}{dt}\]
\( v\) is the velocity of an object (point) on the circle of radius \(R\) and angular position \( \theta \). The circle is on the plane X-Y and the origin is on one point on the circle. - 27angular velocity\[w = \frac{d\theta}{dt}\]
it has units of rad/seg or only \( s^{-1}\) - 28relation between \(v\) and \( w \)\( v = w R\)
- 29velocity as a vector productWhen origin is the O of 3d- plane
\[ \vec{v} = \vec{w} \times \vec{r}\]
where \( R = r sen \theta\) with \( \vec{r}\) a vector position in 3d from O, and \(\theta\) the angle respect the Z-axis.
Eq. valid only for \( \vec{r}\) and \(\theta\) constant (circular motion) - 30Direction of rotationIf the point on the circle is moving counterclockwise \( w \) is perpendicular to the plane of motion following the rule of the right hand closing in the same sense as the motion of the object.
The same for the clockwise. - 31Period Ptime needed to complete one cycle or revolution
- 32frequency \( \nu \)number of revolutions by unit time
- 33Relation between period and frecuency\[ P = \frac{1}{\nu}\]
- 34unitsIf \( P\) is given in seconds
\( \nu\) is given in \( s^{-1} = Hz \) - 35When the concepts of \( P\) and \( \nu\) are validfor all periodic processes which are cyclic.
- 36Example of periodic motion that is not circular not uniform.The motion of Earth around the Sun
- 37For \( w\) constant, circular motion with constant velocity (uniform)\[ \theta = \theta_0 + w (t-t_0)\]
for \( \theta_0 = 0\) at \(t_0 = 0\)
\[ w = \frac{\theta}{t}\] - 38For a complete revolution and \( w \) constant\[ w = \frac{2 \pi}{P}= 2 \pi \nu \]
- 39Circular motion: angular accelerationwhen the angular velocity of an object is changing with time
\[ \vec{\alpha} =\frac{d \vec{w}}{dt}\]
since the motion is in a plane w is orthogonal to this plane but the direction of \(w\) does not change - 40magnitude of angular accelerationthe relation above is true for the magnitude
\[ \alpha = \frac{dw}{dt} = \frac{d^2 \theta}{dt^2}\] - 41for angular acceleration constant (mov. uniformemente acelerado)\[ w = w_0 + \alpha (t-t_0) \]
\[ \theta = \theta_0 + w_0 (t-t_0) + \frac{1}{2} (t-t_0)^2 \]
the last, gives the angular position at any time - 42tangential and normal acceleration in circular motion\[ a_T = \frac{dv}{dt} = R \frac{dw}{dt }= R \alpha\]
\[ a_N = \frac{v^2}{R} = w^2 R\]
Informationen
- Beschreibung
- 1 Motion in one dimension Karteikarten am Curved motion, erstellt von nquirozp am 11/05/2016.
- Erstellt von
- nquirozp
- Aufrufe
- 0
- Bewertung