Physics 131-04 Test 2


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Questions (9 pts. apiece) Answer in complete, well-written sentences WITHIN the spaces provided.

  1. What is the impulse-momentum theorem? Is it correct? What is your evidence?







  2. Suppose the mass of object 1 is much less than that of object 2 and that it is pushing object 2 that has a dead motor so that both objects move in the same direction at speed v.

    \begin{displaymath} m_{1}\ll m_{2}\quad \mbox{and}\quad {{\bf v}_{1}}={{\bf v}_{2}}\end{displaymath}



    \includegraphics{newtons_laws_fig3.eps}



    Predict the relative magnitudes of the forces between object 1 and object 2. Place a check next to your prediction.

    Object 1 exerts more force on object 2.

    The objects exert the same force on each other.

    Object 2 exerts more force on object 1.

    Explain the reasons for your answers.







  3. Suppose a particle is moving around in a circle with an angular velocity that has a magnitude of \( \omega \) associated with it. According to observer #1, does the particle appear to be moving clockwise or counter clockwise? How about the direction of the particle's motion according to observer #2? Is the clockwise vs. counterclockwise designation a good way to determine the direction associated with $\omega$ in an unambiguous way? Why or why not?



    \includegraphics{ang_mom_fig2.eps}(-190,-12)Observer 1 Observer 2




  4. The plot below shows the $x-y$ position of a ball bearing after it was hit by another, rolling ball bearing. The time and positions of the struck bearing are shown in the table. How would you determine the speed of the bearing? Show any equations your would use.

    \includegraphics[height=1.25in]{f1.eps}


    time(s)

    		x (cm) y (cm) time(s) x (cm) y (cm)
    2.5012 22.72 21.568 2.9068 11.52 17.28
    2.5688 22.72 21.568 2.9744 9.004 16.44
    2.6364 22.41 21.464 3.042 6.596 15.71
    2.704 18.85 20.31 3.1096 4.4 15.45
    2.7716 16.54 19.26 3.1772 1.88 13.61
    2.8392 14.13 18.11









  5. The figure below shows four gears that rotate together because of the friction between them so they turn without slipping. Gears 1 and 4 have radius $3R$, gear 2 has radius $R$, and gear 3 has radius $2R$. Gear 2 is forced to rotate by a motor. Rank the four gears according to the angular speed of the gears with the greatest first. Explain your reasoning.

    \includegraphics[height=0.7in]{gears.eps}




Problems. Clearly show all reasoning for full credit. Use a separate sheet to show your work.

1. 15 pts. A steel ball of mass $m_1 = 2.0~kg$ strikes a wall with a speed $v_1 = 10~m/s$ at an angle $\theta = 40^\circ$ with the surface. It bounces off with the same speed and angle. If the ball is in contact with the wall for a time interval $\Delta t = 0.25~s$, what is the average force exerted on the ball by the wall?

\includegraphics[height=1.5in]{bounce.eps}

2. 20 pts. Most of us know intuitively that in a head-on collision between a dump truck and a subcompact car, you are better off being in the truck than in the car. Consider what happens to the two drivers. Suppose each vehicle is initially moving with a speed $v_0 = 10 ~m/s$ and they undergo a perfectly inelastic, head-on collision. Each driver has a mass $m=50~kg$. Including the drivers the total vehicle masses are $m_c=1000~kg$ for the car and $m_t=4000~kg$ for the truck. What is the change in momentum $\Delta \vec p$ for each driver?

3. 20 pts. A student is spinning around on a turntable with her arms outstretched holding a spherical weight in each hand of mass $m_w$ and radius $R_w$. Treat the student's body as a cylinder of mass $m_b$ and radius $r_b$ with thin, massless rods for arms and spheres for the weights on the end of the arms. The left-hand side of the figure below shows a view of her motion from above. The center of each spherical weight is a distance $r_w$ from her axis of rotation. She is spinning initially at a rate of $\omega_0$ and then drops her arms to her side to make a new configuration shown in the right-hand side of the figure. What are her initial and final moments of inertia in terms of the parameters given above? What is her final rotation rate $\omega_1$ in terms of the parameters given above? Ignore the effect of the turntable.


\includegraphics[height=1.5in]{twirling4.eps}

Equations and Constants



\begin{displaymath} \Delta \vec r = \vec r_{finish} - \vec r_{start} \quad \Delta \vec v = \vec v_{finish} - \vec v_{start} \end{displaymath}


\begin{displaymath} \langle \vec v \rangle = {\Delta \vec r \over \Delta t} \qqu... ...ta t \to 0} {\vec r (t+\Delta t) - \vec r(t) \over \Delta t} \end{displaymath}


\begin{displaymath} \langle \vec a \rangle = {\Delta \vec v \over \Delta t} \qqu... ...lta t \to 0} {\vec v(t+\Delta t) - \vec v(t) \over \Delta t} \end{displaymath}


\begin{displaymath} x(t) = {1 \over 2}at^2 + v_0t + y_0 \quad v = at + v_0 \quad... ...er r} \quad ( \vec v \perp \vec r \quad \vec v \perp \vec a_c) \end{displaymath}


\begin{displaymath} \vec F_{net} = \sum_i \vec F_i = m \vec a = {d \vec p \over dt} \qquad \vec F_{AB} = -\vec F_{BA} \end{displaymath}


\begin{displaymath} \vert\vec F_f\vert = \mu N \quad \vert\vec F_c\vert = m {v^2... ...} \quad \vec F_s(x) = -kx\hat i \quad \vec F_g(y) = -mg\hat j \end{displaymath}


\begin{displaymath} W = \int \vec F \cdot d\vec s = \int \vert\vec F\vert ~\ve... ...a U \quad KE = {1 \over 2} mv^2 = {1 \over 2} I \omega^2 \quad \end{displaymath}


\begin{displaymath} U_s(x) = {1 \over 2} kx^2 \quad U_g(y) = mgy \quad U_G = \fr... ... KE_1 + U_1 \quad \vec p = m \vec v \quad \vec p_0 = \vec p_1 \end{displaymath}


\begin{displaymath} \Delta\vec p = \int \vec F dt = \langle \vec F \rangle \Delt... ...er dt} \quad \alpha = {a_T \over r} = {d\omega \over dt} \quad \end{displaymath}


\begin{displaymath} I = \sum m_i r_i^2 = I_{cm} + MD^2 \quad \vert\vec \tau \v... ...ha \quad L = I \omega \quad L_0 = L_1 \quad v_{cm} = r\omega \end{displaymath}


\begin{displaymath} \vec A = A_x \hat i + A_y \hat j + A_z \hat k \qquad {dA \ov... ... 0 \qquad {dt \over dt} = 1 \qquad {dt^2 \over dt} = 2t \qquad \end{displaymath}


\begin{displaymath} \int f(x)dx = \lim_{\Delta x \rightarrow 0} \sum f(x_i) \D... ... x \qquad \int dx = x + c \qquad \int x dx = {x^2 \over 2} + c \end{displaymath}


\begin{displaymath} \sin \theta = {opp \over hyp} \quad \cos \theta = {adj \over... ...over adj} \quad x^2 + y^2 + z^2 = R^2 \quad \rho = {m \over V} \end{displaymath}


\begin{displaymath} x = {-b \pm \sqrt{b^2 -4ac} \over 2a} \quad C = 2 \pi r \qua... ...pi r^2 \quad Area = {1 \over 2}bh \quad Area = 4 \pi r^2 \quad \end{displaymath}

Speed of Light ($c$) $2.9979\times 10^8~m/s$ proton/neutron mass $1.67\times 10^{-27}~kg$
$R$ $8.31J/K-mole$ $g$ $9.8~m/s^2$
Gravitation constant $6.67 \times 10^{-11}~N-m^2/kg^2$ Earth's radius $6.37\times 10^6~m$
Earth-Moon distance $3.84\times 10^8~m$ Electron mass $9.11\times 10^{-31}~kg$

\includegraphics[width=6.0in]{10t02a.ps}
\includegraphics[width=6.0in]{10t02b.ps}