Physics 131-04 Test 2

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

1. Newton's Second Law states that $\vec F = m \vec a$. Why should you believe this? What is your evidence?

   
   
   
   
   
   
   

2. A cart is moving toward the right and slowing down, as shown in the diagrams below. Draw arrows above the cart representing the magnitudes and directions of the net (combined) forces you think are needed on the cart at $t = 0$ s, $t = 1$ s, etc. to maintain its motion with a steadily decreasing velocity.

   
   
   

   
   
   
   

   Explain the reasons for your answers.

   
   
   
   
   
   
   

3. Take a look at the diagram below. Can the strongest member of your group stretch a string or rope so that it is perfectly horizontal when a 10 kg mass is hanging from it? In other words, can the string provide a force that just balances the force exerted by the mass? Explain.

   
   
   
   
   
   
   
   

4. Recall the lab where we used toy airplanes to study centripetal force. In that lab we used video analysis to measure $r$, the radius of the airplane's circular path, and a ruler to measure $R$, the total length of the string that is actually rotating below the pivot point. Using these two distances ($r$ and $R$), calculate the angle the string makes with the horizontal. Clearly show your reasoning.

   
   
   
   
   
   
   

5. Consider the result below for a measurement of the kinetic, potential, and total energy of a falling ball. Does the mechanical energy appear to be conserved within experimental uncertainties? How would you quantitatively estimate the value of the experimental uncertainty? Once you establish the method apply it to your data. The average value of the total energy in the figure is $0.63\pm 0.05~J$.

   

   
   
   
   

6. The figure below shows four situations - one in which an initially stationary block is dropped from rest (part (1) on the left-hand side of the figure below) and three in which the block is allowed to slide down frictionless ramps (parts (2)-(4) in the figure). Rank the situations according to the speed of the block at point $B$ with the greatest first. Explain your reasoning.

   
   
   
   
   

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

1.	15 pts.	An object with mass $m_1 = 4~kg$ is placed on a frictionless, horizontal table and connected to a cable that passes over a pulley and then is fastened to another object with $m_2 = 10~kg$ as shown below. What is the acceleration of each object and what is the tension in the string?
2.	17 pts.	A block of mass $m = 1.0~kg$ is pushed against a horizontal spring of negligible mass until the spring is compressed a distance $x$. See the figure below. The force constant of the spring is $k=1000~N/m$. When l3.0in the spring is released the block travels along a frictionless, horizontal surface to point $B$, the bottom of a vertical circular track of radius $R = 1.0~m$, and continues to move up the track. The speed of the block at the bottom of the track is $v_b = 10~m/s$ and the block experiences an average friction force $F_f = 5~N$ while sliding up the track. What is $x$? What is the speed $v_T$ of the block at the top of the track? Assume it has enough speed to make it to the top.
3.	20 pts.	The space shuttle is in a high, circular orbit a distance $h_1$ above the surface of the Earth. The crew performs an experiment that needs to take place far away from the spacecraft so a payload of mass $m_1$ is `lowered' toward the Earth on a massless rope of length $h_2$. The shuttle and the payload are flying along in equilibrium so the spacecraft, the payload, and the rope are all aligned along a radius from the center of the Earth to the shuttle. The period of the shuttle's orbit $T_s$ is known, the mass $m_1$ of the payload is much smaller than the shuttle mass and $h_2 << h_1$. How is the shuttle's period $T_s$ related to the payload's period $T_p$? What is the tension $F_{rope}$ in the rope in terms of $T_s$, $h_1$, $h_2$, $m_1$ and any other known constants? Be careful with the notation because we sometimes use $T$ to refer to tension and the period of the orbit.

Physics 131-4 Exam Sheet, Test 2

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

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

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

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

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

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

$$W = \int \vec F \cdot d\vec s = \int \vert\vec F\vert \ver... ...vert \cos\theta = \Delta KE \quad KE = {1 \over 2} mv^2 \quad$$

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

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

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

$$\int f(x)dx = \lim_{\Delta x \rightarrow 0} \sum f(x_i) \Delta x \qquad$$

$$\sin \theta = {opp \over hyp} \quad \cos \theta = {adj \over... ...theta_2 + \cos\theta_1\sin\theta_2 \quad x^2 + y^2 + z^2 = R^2$$

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

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

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	$7.3477\times 10^{22}~kg$