Physics 131-01 Test 3

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Questions (8 pts. apiece) Answer in complete, well-written sentences WITHIN the spaces provided.

1. Suppose a ball of mass 0.20 kg is dropped and falls toward the surface of the moon so that it hits the ground with a speed of 40 m/s and rebounds with the same speed. Consider the ball and the moon as an interacting system with no other outside forces. Why might the astronaut (who hasn't taken physics yet!) have the illusion that momentum isn't conserved in the interaction between the ball and the moon?

   
   
   
   
   
   
   

2. Using the diagram below, draw a dotted line in the direction you think your two cars will move after a perfectly elastic collision between cars with different masses and the same velocities. Explain your reasoning in the space below.

   

   
   
   
   

3. Assume that an object is moving in a circle of constant radius, $r$ and recall that $s=r\theta$ where $s$ is the arc length along the edge of a circle, $r$ is the radius of the circle, and $\theta$ the angle. Take the derivative of $s$ with respect to time to find the velocity of the object. Show that the magnitude of the linear velocity, $v$, is related to the magnitude of the angular velocity, , by the equation .

   
   
   
   
   
   
   
   

4. Recall the laboratory where you studied a rotational collision. You dropped from rest a weight onto a spinning rotator and measured the motion before and after dropping the weight from rest to test the conservation of angular momentum. Would the procedure you followed change if the weight was moving horizontally at a constant velocity when you dropped it? If it changed, what would be different?

   
   
   
   
   
   
   
   

5. As global warming continues over the next century, some of the polar ice fields will melt and more water will end up near the equator. How would this change the moment of inertia of the Earth? Would the length of a day increase or decrease? Explain.

   
   
   
   
   
   
   
   

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

1.

15 pts.

As shown in the figure, a bullet of mass $m_b$ and speed $v$ passes completely through a pendulum bob of mass $M_p$. The bullet emerges with a speed $v/2$. The pendulum bob is suspended by a stiff rod of length $l$ and negligible mass. What is the minimum value of $v$ such that the pendulum bob will barely swing through a complete vertical circle? Your answer should be in terms of $M_p$, $v$, $m_b$, and $l$ and any other necessary constants.

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

2.	20 pts.	Measuring the moment of inertia of an irregularly-shaped object like the payload of a spacecraft can be done with a device like the one shown in the figure. A counterweight of mass $m$ is suspended by a cord wound around a spool of radius $r$, forming part of a turntable supporting the object. The turntable can rotate without friction. When the counterweight is released from rest, it descends a distance $h$, acquiring a speed $v$. Show that the moment of inertia $I$ of the rotating apparatus including the turntable is $mr^2(2gh/v^2 - 1)$.
3.	25 pts.	A widely accepted theory of planet formation states that planets form out of dust that collides and sticks to form larger and larger bodies. When two of these `planetesimals' hit they can coalesce and form (through melting from the heat released in the collision) a single, spherical object. A non-spinning planetesimal of mass $m$ and radius $r$ is moving with velocity $\vec v_1$ to the right as shown in the figure below. It collides with an identical (same $m$ and $r$), non-spinning planetesimal moving to the left with velocity $\vec v_2$ and they stick together. The velocities are parallel, but in opposite directions. Their initial trajectories are a perpendicular distance $d$ apart as shown in the figure. They eventually form a uniform spherical body with mass $2m$ and radius $1.260r$. What is the velocity $v_{cm}$ of the final object after the collision in terms of the information given above ($m$, $r$, $\vec v_1$, $\vec v_2$, and $d$)? What is its angular velocity $\omega$ after the collision in terms of the same information?

Physics 131-1 Exam Sheet, Test 3

$$\Delta \vec r = \vec r_{finish} - \vec r_{start} \quad \Delta \vec v = \vec v_{finish} - \vec v_{start}$$

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

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

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

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

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

$$W = \int \vec F \cdot d\vec s = \int \vert\vec F\vert ~\ve... ...} = {1 \over 2} I \omega^2 \quad KE_0 + U_0 = KE_1 + U_1 \quad$$

$$U_s(x) = {1 \over 2} kx^2 \quad U_g(y) = mgy \quad U_G = \fr... ...vec p_1 \quad \vec r_{cm} = \frac{\sum \vec r_i m_i}{\sum m_i}$$

$$\theta = {s \over r} \quad \omega = {v_T \over r} = {d\theta... ...ega \over dt} \quad I = \sum m_i r_i^2 = I_{cm} + Mh^2 \quad$$

$$\vert\vec \tau \vert = r F\sin\phi = I\alpha = \left \vert\f... ...mv\sin \phi = I \omega \quad L_0 = L_1 \quad v_{cm} = r\omega$$

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

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

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

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

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	$9.11\times 10^{-31}~kg$