Physics 131-04 Final Exam

I pledge that I have neither given nor received unauthorized assistance during the completion of this work.

Signature height0pt depth1pt width3in

Questions (5 pts. apiece) Answer in complete, well-written sentences WITHIN the spaces provided.

1. Consider the following scenario from the Twins Paradox of Special Relativity. As a space-faring twin's craft recedes from the Earth it is moving at a constant speed. Since no inertial frame can be considered ``better'' than any other there is nothing physically inconsistent with the view that the space-faring twin is observing the Earth recede from her at a constant velocity. Hence, the space-faring twin will observe clocks on the Earth to move slowly and the Earth-bound twin will age at a slower rate than the space-faring one. Is this reasoning flawed? Explain.
   
   
   
   
   
   
   

2. A spacecraft has the shape of a sphere when viewed from an inertial frame at rest with respect to the spacecraft. If it moves past an observer on Earth with a speed of $v=0.5c$, what shape does the observer measure for the spacecraft? Explain.
   
   
   
   
   
   
   

3. In the bottom panel of the figure below sketch the acceleration versus time graph that would match the velocity versus time plot that is shown in the top panel of the figure. State the reasoning behind your sketch.
   
   

   

   
   
   

4. Recall the lab where you placed a motion sensor at the bottom of a dynamics track inclined upward and you gave a dynamics cart a gentle push away from the motion detector. For each part of the motion - away from the detector, at the turning point, and toward the detector, predict in the table below whether the velocity will be positive, zero or negative. Also indicate whether the acceleration is positive, zero or negative. Explain your reasoning.

   
   

   	Moving Away	Turning Around	Moving Toward
   Velocity
   Acceleration

   
   
   
   
   
   
   
   

5. Consider the histogram shown below for two sets of possible fall times from different classes. The average and standard deviation of the data is $0.73\pm 0.40 ~ s$. How many peaks are in this histogram? Do the average and standard deviation capture the full description of the data? Why or why not.

   

   
   
   

6. For the configuration shown below, predict the reading in newtons on the spring scale ($m=1.0~kg$). Explain your reasoning.
   

   
   
   

7. You have a spring scale hooked to a block on a flat, lab table. Hold the spring scale at a 30 angle with respect to the horizontal and use it to pull the block a distance of 0.5 meters along the table surface so the block moves at a constant speed. Is this force larger or smaller than if you pulled the block horizontally? Explain.
   
   
   
   
   
   
   
   

8. The figure below shows for graphs of position versus time for two bodies and their center of mass. The two bodies undergo a completely inelastic, one-dimensional collision while moving along the $x$ axis. Which graphs correspond to physically impossible situations. Explain your reasoning.
   

9. The figures below show identical meter sticks, half wood and half steel, that are pivoted at the point $O$ in figure (a) and at the point $O^\prime$ in figure (b). The same force $\vec F$ is applied at the points $a$ and $a^\prime$ in figures (a) and (b) respectively. The steel portion of the meter stick is heavier than the wood part. Is the angular acceleration in the figure (b) the same, less than, or greater than the angular acceleration in figure (a)? Explain.
   
   
   
   
   
   
   
   
   

10. A skater slides down three slopes of frictionless ice whose vertical heights $d$ are identical. Rank the slopes according to change in her kinetic energy produced along the slope, greatest first. Explain your reasoning.
    

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

1.	8 pts.	An interstellar space probe is launched from the Earth. After a brief period of acceleration it moves with a constant velocity $v = 0.7c$ where $c$ is the speed of light. Its nuclear-powered batteries supply the energy to keep its data transmitter active continuously. The batteries have a lifetime $\Delta t_b = 15~yr$ as measured in a rest frame. How long do the batteries last as measured by Mission Control on Earth? How far is the probe from the Earth when its batteries fail as measured by Mission Control?
2.	8 pts.	A man is reported to have fallen a distance $h_0=50~m$ from a building, landing on a metal ventilator box that he crushed to a depth $h_1=0.5~m$. He suffered only minor injuries. Ignoring air friction and treating his acceleration after hitting box as constant, how long did it take for him to crush the box after he landed on it? Assume he fell from rest.
3.	8 pts.	An object of mass $m=1~kg$ is observed to accelerate at $a=10~m/s^2$ in a direction $\theta = 40^\circ$ north of east as shown in the figure. The force $\vec F_2$ acting on the object has a magnitude $\vert\vec F_2\vert = 6.4~N$ and is directed north. What is the magnitude and direction of the force $\vec F_1$?

Continue $\rightarrow$

Problems (continued). Clearly show all work for full credit.

4.	8 pts.	It takes $W_0 = 5.0~J$ of work to stretch a Hooke's-Law spring a distance $x=0.1~m$ from its unstressed length. What is the extra work $W_1$ required to stretch it an additional $0.12~m$?
5.	8 pts.	To measure the speed of a bullet, the following situation is set up. A wooden block ($m_w=0.50~kg$) is put on top of a fencepost with a height of $h=1.5~m$. The bullet ($m_b=0.010~kg$) strikes the block from a perfectly horizontal direction and remains embedded in it. The block is measured to fall $x_1=1.6~m$ from the base of the post. How fast was the bullet going?
6.	10 pts.	One of the components of an old grandfather clock consists of a uniform ring attached to a pivot by a short, uniform rod. It starts from a nearly vertical position as shown in the figure and is then released from rest. What is the speed of the lowest point on the hoop at the bottom of the hoop's trajectory? The mass and radius of the hoop are $m_h$ and $R_h$. The length of the rod is $l_r$ and the mass is so small that we can ignore it. Get your answer in terms of these quantities and any other necessary constants.

Physics 131-4 Final Exam Sheet

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

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

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

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

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

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

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

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

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

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

$$\Delta t = {\Delta t_p \over \sqrt {1 - {v^2 \over c^2}}} \q... ...^\prime = x - vt \quad y^\prime = y \quad v_i^\prime = v_i - v$$

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

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

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

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

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