PHYS 218 Exam 3

Energy and Momentum

Momentum: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P = mv} Impulse: $J=\int _{t_{i}}^{t_{f}}F\ \mathrm {d} t=P_{f}-P_{i}$

Collisions

Conservation of Momentum

Elastic: (All energy conserved: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_i = K_f} )

Inelastic: (Max loss of energy: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_f = K_i} ) When two bodies stick together.

Center of Mass

Position: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_{cm} = \frac{\sum m_i r_i}{\sum m_i}}

Velocity: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_{cm} = \frac{\sum m_i v_i}{\sum m_i}}

Acceleration: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{cm} = \frac{\sum m_i a_i}{\sum m_i}}

In a system

Sum of all external forces is the total mass × a_cm = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{dP_{cm}}{dt}}

If sum of forces in any component direction is 0, the energy is conserved in that direction

Rotational Kinematics

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \omega &= \frac{\mathrm{d}\theta}{\mathrm{d}t} \\ \alpha &= \frac{\mathrm{d}\omega}{\mathrm{d}t} = \frac{\mathrm{d}^2\theta}{\mathrm{d}t^2} \\ \omega &= \omega_0 + \alpha t \\ \theta - \theta_0 &= \omega_0t + \frac{1}{2} \alpha t^2 \\ v &= R\omega \\ a_T &= R\alpha \\ a_R &= R\omega^2 \end{align}}

Moment of Inertia

With respect to an axis Q:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_Q = \sum m_i r_{iQ}^2}

Parallel Axis Theorem

Rotate an object (Moment of Inertia for center of mass is known) around a new axis at a point Q)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_Q = I_{cm} + Md^2}

Angular Momentum

Conversion from linear Momentum: Sum of the cross product of position and linear momentum for each particle

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_Q = \sum r_{iQ} \times m_iv_i}

Object rotating around its center of mass:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L = I\omega}

Rotational Energy

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K = \frac{1}{2} I \omega^2}

Torque

Cross product between position applied and force:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau = r \times F}

Sum of all torques is equivalent to moment of inertia times the rotational acceleration.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum \tau = I \alpha = \frac{\mathrm{d}L}{\mathrm{d}t}}

If the sum of all torques is equal to 0, then we have conservation of angular momentum.

Precession

occurs when angular momentum L is perpendicular to a torque. A rotating object will move around an axis parallel to the applied force:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega = \frac{\tau}{L}}

Topics for Exam

1. Angular Momentum
   • When it is conserved an when it is not
   • Conservation of momentum and parallel axis theorem
2. Rotational Kinematics (falling yo-yo)
   • Use conservation of energy
   • Use kinematics equations (sum of forces, sum of torques)
3. Equilibrium Problems
   • Find tension (make sure that sum of forces and sum of torques are zero from any point)
4. Energy and Collisions
   • Is it elastic or inelastic? Why?