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A modeling approach to the 8.01 final exam equation sheet.
Interactions
General Definitions
Work
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^
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\vec
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\cdot d\vec
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]\end
Potential Energy
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Unknown macro: {large} [\Delta U = -W_
Unknown macro: {rm conservative}
= -\int_
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^
Unknown macro: {B}
\vec
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_
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\cdot d\vec
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]\end
Power
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\cdot\vec
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\qquad \qquad P_
Unknown macro: {rot}
=\frac{dW_{rot}}
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= \tau_
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\omega]\end
Impulse
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Unknown macro: {large} [ I = \int_
Unknown macro: {t=0}
^
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\vec
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(t)\:dt ]\end
Torque
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= \vec
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_
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\times\vec
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_
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\qquad \qquad |\vec
_
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| = |\vec
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_
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||\vec
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_
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| \sin\theta = r_
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]\end
Net Torque
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_
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= \vec
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_
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\times \sum_
\vec
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+ \sum_
\vec
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_
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\times \vec
_
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^
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]\end
Specific Interactions
Gravity – Universal
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Unknown macro: {large} [ \vec
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_
Unknown macro: {12} = - G\frac{m_
Unknown macro: {1}
m_{2}}{r_
^{2}}\hat
Unknown macro: {r}
_
Unknown macro: {12} \qquad\qquad U_
(r) = - G\frac{m_
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m_{2}}{r_{12}}]\end
Gravity – Near Earth
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Unknown macro: {large} [ F = mg \mbox
Unknown macro: { (directed straight downward)}
\qquad \qquad U
= mgy ]\end
Contact Force
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_
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= \vec
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+ \vec
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]\end
Friction – Static
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Unknown macro: {s} \le f_
Unknown macro: {s,max}
= \mu_
N \mbox
Unknown macro: { (directed opposite net force neglecting friction)}
]\end
Friction – Kinetic
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Unknown macro: {large} [ f_
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N \mbox
Unknown macro: { (opposes motion with respect to the surface)}
]\end
Springs
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Unknown macro: {large} [ F = k|\Delta x| \mbox
Unknown macro: { (restoring)}
\qquad \qquad U
= \frac
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]\end
System Characteristics
Center of Mass
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_
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= \frac
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^
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m_
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_
\rightarrow \frac
{m^{total}} \int_
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dm\: \vec
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]\end
Velocity of Center of Mass
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_
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= \frac
Unknown macro: {1} {m^{total}} \sum_
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^
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m_
Unknown macro: {i} \vec
Unknown macro: {v}
_
\rightarrow \frac
{m^{total}} \int_
Unknown macro: {body}
dm\: \vec
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]\end
Momentum
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\qquad \qquad \vec
^
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= \sum_
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^
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m_
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_
]\end
Translational Kinetic Energy
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\qquad \qquad \Delta K = \frac
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^
- \frac
Unknown macro: {1}
Unknown macro: {2} mv_
Unknown macro: {0}
^
]\end
Period
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Unknown macro: {2pi}
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]\end
Frequency
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Unknown macro: {large} [ f = \frac
Unknown macro: {1}
Unknown macro: {T}
= \frac
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]\end
Moment of Inertia
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dm\:r_
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^
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\qquad\qquad I_
=md^
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+ I_
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\mbox
Unknown macro: { (Parallel Axis Theorem)}
]\end
Rotational Kinetic Energy
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Unknown macro: {large} [ K = \frac
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]\end
Total Kinetic Energy
Unknown macro: {latex} \begin
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Unknown macro: {total} = \frac
Unknown macro: {1}
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m^
v_
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I_
\omega^
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]\end
Angular Momentum (Point Particles)
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_
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= \sum_
Unknown macro: {i} \vec
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_
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\times m_
\vec
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_
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]\end
Angular Momentum (System)
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_
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= \vec
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_
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\times m^
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\vec
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_
\vec
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]\end
Models
Momentum and Impulse
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_
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^
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= \frac{d\vec
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\qquad \qquad \Delta \vec
= \vec
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]\end