PHYS 218 Chapter 8
« previous | Monday, March 7, 2011 | next »
Linear Momentum and Impulse
From Newton's laws, we know that
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_{obj B}\mathbf{F} = m\mathbf{a} = m\frac{d\mathbf{v}}{dt} = \frac{d(m\mathbf{v})}{dt} = \frac{d\mathbf{P}}{dt}}
- Linear Momentum of object
- mass × 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 \mathbf{P} = m\mathbf{v}}
- Impulse of object
- change in momentum (final − initial)
- 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 \mathbf{J} = \int_{t_i}^{t_f} \sum_{obj B}\mathbf{F}\ dt = \mathbf{P}_f-\mathbf{P}_i}
- if force is constant: 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 \mathbf{J} = \mathbf{F} \cdot \Delta t}
- Note: 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 \mathbf{F}_{avg} = \frac{\mathbf{J}}{\Delta{t}}} .
Example
A "complicated apparatus" takes a 1 kg object moving into it at 20 m/s and spits it out 0.1 s later at an angle 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 \varphi} at 30 m/s. What is the impulse?
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} \mathbf{J}&=\mathbf{P}_f-\mathbf{P}_i \\ &= (30\cos{\varphi}-20)\hat{\mathbf{x}}+30\sin{\varphi}\hat{\mathbf{y}} \end{align}}
Average 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 \frac{\mathbf{J}}{0.1}}
Conservation of Momentum
By Newton's 3rd law, we know that every force has an equal and opposite force (e.g. normal). Since momentum is the change of force over time 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 \left(\textstyle\sum \mathbf{F} = \tfrac{d\mathbf{P}}{dt}\right)} , the overall system's momentum is conserved.
Sum of forces over system: 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{\mathbf{F}} = \frac{d}{dt} \left(\sum_{\mathrm{particles}} m_i \mathbf{v}_i\right) = \frac{d}{dt}\left(\sum \mathbf{P}_i\right)}
If the forces in a particular direction are not conserved (i.e. gravity), conservation of momentum may still apply in the other components
Example 1
Before: two equal masses are moving toward each other with equal velocities.
After: the masses collide and stick together
- Initial 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 2\times \frac{1}{2}mv^2 = mv^2}
- Final Energy
- no movement, so 0
Clearly, there is no conservation of energy (Initial energy is greater than 0, while the final energy is 0)
- Work of non-conservative forces is done by internal forces (i.e. deforming the objects as they collide)
However, momentum is conserved since there are no external forces and 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_i = P_f} .
- 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_i = mv_1 + mv_2 = mv_1-mv_1 = 0}
- 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_f = 0}
Example 2: Problem 8.18
When cars are equipped with flexible bumpers, they will bounce off each other during low-speed collisions, thus causing less damage. In one such accident, a 1750-kg car traveling to the right at 1.5 m/s collides with a 1450-kg car going to the left at 1.1 m/s. Measurements show that the heavier car's speed just after collision was 0.25 m/s in its original direction. You can ignore any road friction during the collision.
- What was the speed of the lighter car just after the collision?
Calculate the change in the combined kinetic energy of the two-car system during this collision.
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} P_i &= m_1v_1 + m_2v_2 = 1750 \cdot 1.5 - 1450 \cdot 1.1 \\ P_f &= m_1v_1' + m_2v_2' = 1750 \cdot 0.25 + 1450v_2' \\ v_2' &= 0.408\ \mbox{to the right} \end{align}}
Collisions
"Strong interactions between bodies that occur in a short amount of time"
- elastic: all kinetic energy is conserved
- inelastic: final kinetic energy is less than the initial kinetic energy (energy transformed into internal energy)
- completely inelastic: maximal loss of energy, but still maintains conservation of momentum (usually the case where objects stick together)
Center of Mass
Wednesday, March 23, 2011
position of the 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 \mathbf{r}_{cm}} ) is the "mass-weighted average position"
Weighted average: 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 \bar{x} = \frac{\sum{w_ix_i}}{\sum{w_i}}}
For a system of continuum particles (like a solid object):
where 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 \rho(\mathbf{r})} is the density at a given point and V is the volume
If the density is homogeneous, then
Example
Particle of mass 1 kg at (1, 2) and another particle of mass 2 kg at (3, 1).
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} \mathbf{r}_{cm} &= \frac{1 \langle 1,2 \rangle + 2 \langle 3,1 \rangle }{3} \\ &= \left\langle \frac{7}{3}, \frac{4}{3} \right\rangle \end{align}}
Motion of the Center of Mass
Monday, March 28, 2011
Velocity is derivative of position; can also be obtained from individual particles. Same with acceleration.
Forces on a System
From the acceleration of the center of mass above, we can solve the following given 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 M=\sum{m_i}} (sum of all masses):
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 M\mathbf{a}_{cm} = \sum{F_i}}
This equation gives us the value of the external forces, which happens to be related to the change in momentum of the system:
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 \mathbf{F}_{ext} = M\mathbf{a}_{cm} = M \frac{d\mathbf{p}}{dt}}
Center of mass of a system moves just as though all masses were concentrated at the center of mass and acted upon by a net external force.
Therefore, momentum is conserved iff the sum of all external forces is 0