Physics 131-04 Test 2

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

1. 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. Assume that friction is so small that it can be ignored. Explain the reasons for your answers.

   
   
   
   
   

   
   
   
   
   

2. Suppose you were to hang equal masses of $m = 0.5~kg$ in the configuration shown below with the spring scale. Predict the tension in the string and the reading on the spring scale. Explain your reasoning.

   
   
   
   
   

3. Is the gravitational acceleration ``constant'', $g$, really a constant? Explain.
   
   

4. Consider a ball dropped from rest near the Earth's surface with $y_0=0$ and positive is down. Using the definition of work, show that $W = mgy$ when an object is dropped through a distance $y$. Clearly show your reasoning.

   
   
   
   
   
   
   
   

5. The velocity $v$ of the falling object in the previous question is related to the position $y$ by $v=\sqrt{2 y g}$. Combine this equation with the result in question 4 to show in theory the work done on a mass falling under the influence of the gravitational attraction exerted on it by the earth is given by the work-kinetic-energy theorem. Clearly show your reasoning.

   
   
   
   
   
   
   
   

6. Consider the figure below showing the potential energy $U(x)$ of a one-dimensional system as a function of position. What is the maximum value of the mechanical energy $E$ if the particle is to be trapped in the potential well, i.e., the region $A<x<D$? Explain your reasoning.

   
   
   
   
   

   
   
   
   
   
   
   

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

1.	12 pts.	How fast can you set the Earth moving? In particular, when you jump straight up as high as you can, what is the maximum recoil speed that you give to the Earth? Assume you have a mass $m=80~kg$ and can raise your center of gravity by a distance $y=0.20~m$. Your initial speed $v_0$ is related to the final height by $v_0 = \sqrt{2y g}$.
2.	15 pts.	The figure below shows what is known as a conical pendulum. The mass $m$, attached to a string, moves in a horizontal circle of radius $r$, with tangential velocity $v$. What is the angle that the string makes with the axis of the cone that the pendulum sweeps out in terms of $m$, $r$, $v$, and any additional constants?
3.	25 pts.	A new event has been proposed for the Winter Olympics. An athlete will sprint a distance $x_1 = 100~m$, starting from rest, and leap into a bobsled of mass $m_b=20~kg$. The person and the bobsled will slide down a ice-covered ramp of length $l_r=50~m$ and tilted at an angle $\theta_r=20^\circ$ to the horizontal. At the bottom of the slope awaits a spring with spring constant $k=2000~N/m$. The athlete who compresses the spring the most wins. The US entry in the competition Lisa has a mass $m_L = 40~kg$ and she can reach a maximum velocity $v_L = 12 ~m/s$ in the 100-m dash. How far will Lisa compress the spring? Ignore any friction with the ice.

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