Physics 131-01 Test 2

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

1. Newton's Third Law states that $\vec F_{AB} = \vec F_{BA}$ where the subscript $AB$ refers to the force on object $B$ due to object $A$ and $BA$ is the opposite. Why should you believe this? What is your evidence?

   
   
   
   
   

2. A dynamics cart, pulley, hanging mass, and motion detector are set up as shown below.

   
   

   

   Suppose you position the cart 0.10 m from the motion detector and give it a push away from the motion detector and release it. Draw below vectors which might represent the velocity, force and acceleration of the cart at each time after it is released and is moving toward the right. Be sure to mark your arrows with , , or as appropriate. Explain your reasoning.

   

   
   
   

3. Start with Newton's law of universal gravitation to show that the magnitude of the acceleration due to gravity on an object of mass $m$ at a height $h$ above the surface of the earth is given by the following expression ($M_e$ and $R_e$ are the Earth's mass and radius).
   
   $$\frac{GM_{e}}{\left( R_{e}+h\right) ^{2}} \qquad\qquad \qquad\qquad \qquad\qquad \qquad\qquad \qquad\qquad$$
   
   

   
   

   $\parbox{3.0in}{Hint: Because of the spherical symmetry of the Earth ... ... of the Earth as if it were all concentrated at a point at the Earth's center.}$

   
   
   
   
   
   
   
   

4. In the Centripetal Force lab you extracted the centripetal force $F_c$ on the toy airplane using the mass $m$, average speed $v$, radius $r$ of the circular motion, and $F_c = mv^2/r$. You then determined the angle $\theta$ of the string to the horizontal using $r$ and $R$ the total length of the string that is actually rotating. Now generate an expression for the vertical component of the force exerted by the string. Make a vector diagram of the different components. Generate an expression for the total force acting on the airplane.
   
   
   
   

   
   
   
   
   

5. When a particle moves from $f$ to $i$ and from $j$ to $i$ along the paths shown in the figure and in the directions shown by the arrows along each trajectory, a conservative force $\vec F$ does the indicated amounts of work on it. How much work is done by $\vec F$ when the particle moves from $f$ to $j$? Explain.

   

   
   
   
   
   
   
   

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

1.	15 pts.	Two blocks of mass $M$ and $3M$ are placed on a horizontal, frictionless surface. A light spring is attached to one of them and the blocks are pushed together with the spring between them (see figure). A cord initially holding the blocks together is burned; after this the block of mass $3M$ moves to the right with a speed $v_1 = 3.0~m/s$. What is the speed of the block of mass $M$? What is the original elastic potential energy of the spring if $M=0.5~kg$?
2.	20 pts.	A simple accelerometer is constructed inside a car by suspending an object of mass $m$ from a string of length $L$ that is tied to the car's ceiling. As the car accelerates the car-string system makes a constant angle of $\theta$ with respect to the vertical. If the string mass is negligible, then derive an expression for the car's acceleration in terms of $\theta$ and show that it is independent of the mass $m$ and the length $L$.
3.	25 pts.	The ball launcher in a classic pinball machine has a spring with a force constant $k=120~N/m$ (see figure). The surface on which the ball moves is inclined at $\theta = 10^\circ$ with respect to the horizontal. The spring is initially compressed $d = 0.05~m$. What is the launching speed of a ball of mass $m=0.1~kg$ when the plunger is released? Friction and the mass of the plunger are negligible.

Physics 131-1 Exam Sheet, Test 2

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

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

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

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

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

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

$$W = \int \vec F \cdot d\vec s = \int \vert\vec F\vert \ver... ...mv^2 \quad \vec I = \int_{t_1}^{t_2} \vec F dt = \Delta \vec p$$

$$PE_g = mgh \quad PE_s = {1\over 2}kx^2 \quad PE_G = -\frac{G m_1 m_2}{r}$$

$$ME = KE + PE \quad ME_f = ME_i \rightarrow KE_f + PE_f = KE_i + PE_i \quad \vec p = m\vec v \rightarrow \vec p_i = \vec p_f$$

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

$$\int f(x)dx = \lim_{\Delta x \rightarrow 0} \sum f(x_i) \Delta x \qquad \int \frac{1}{x^2} dx = -\frac{1}{x} + C$$

$$\sin \theta = {opp \over hyp} \quad \cos \theta = {adj \over... ...uad \tan \theta = {opp \over adj} \qquad x^2 + y^2 + z^2 = R^2$$

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

$$V = {4 \over 3}\pi r^3 \quad V = \pi r^2 ~l \quad \theta = {s \over r} \quad \rho = {m \over V}$$

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	$3.84\times 10^8~m$	Earth mass	$5.9742\times 10^{24}~kg$
Electron mass	$9.11\times 10^{-31}~kg$	Moon mass	$7.3477\times 10^{22}~kg$