Physics 131-4 Final Exam

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Questions (4 pts. apiece) Answer in complete, well-written sentences WITHIN the spaces provided.

1. You are riding on a flat surface in a cart at a velocity $\vec v_{cart}$. You point a toy cannon straight up in the air, fire off a shot, and record the vertical position as a function of time. An observer on the ground also measures the vertical position as a function of time. How do the two plots differ in appearance? Explain your answer.

2. Consider the following scenario from the twins paradox. As the 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? Why or why not?

3. The velocity graph below shows the motion of two objects, A and B. Answer the following questions separately. Explain your answers. (1) What does the intersection mean? (2) Can one tell which object is ``ahead''? Explain your reasoning. (define ``ahead'')

   
   

4. Consider the histogram shown below for two sets of possible fall times from different classes. The average and standard deviation of the data is $0.95\pm 0.36 ~ s$. Do the average and standard deviation capture the full description of the data? Why or why not?

   
   

   

5. Use 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.
   
   $$\frac{GM_{e}}{\left( R_{e}+h\right) ^{2}}$$
   
   

   Hint: Because of the spherical symmetry of the Earth you can treat the mass of the Earth as if it were all concentrated at a point at the Earth's center.

6. Assume that an object is accelerating in a circle of constant radius, $r$. Using the relationship $v=r\omega$, take the derivative of $v$ with respect to time to find the tangential acceleration of the object. Show that the linear acceleration, , tangent to the circle is related to the angular acceleration, , by the equation .

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

   
   
   

   

   
   
   

8. Suppose an object is lifted through a distance $s$ near the surface of the Earth and then allowed to fall. Starting from the definition of work as $W = \vec F \cdot \vec s$ show $W = mgy$ when the object is dropped through a distance $y$. Explain your reasoning.

9. Suppose somebody tosses you a raw egg and you catch it. In physics jargon, one would say (in a very official tone of voice) that ``the egg and the hand have undergone an inelastic collision.'' What is the relationship between the force you have to exert on the egg to stop it, the time it takes you to stop it, and the momentum change that the egg experiences? You ought to have some intuition about this matter. In more ordinary language, would you catch an egg slowly or fast?

10. The figure below shows the velocity and acceleration of a particular instant in three situations. In which situations is the speed not changing. Explain.

    

11. The figure below shows an overhead view of two, identical stones traveling in circles over a frictionless surface. Each stone is tied to a cord whose opposite end is anchored at the center of the circle. Is the tension in the longer cord greater than, less than, or the same as the tension in the shorter cord if the stones travel with the same period of motion? Explain.

    

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

1.	8 pts.	An astronomer on the Earth observes a meteoroid in the southern sky approaching the Earth at a speed $v_0 = 0.85c$. At the time of its discovery the meteoroid is a distance $x_0 = 1.0~ly$ from Earth. What is the time interval for the meteoroid to reach the Earth as measured by the Earthbound astronomer? What is the time interval as measured by an alien tourist traveling on the meteoroid?
2.	8 pts.	A car of mass $m_c = 1400~kg$, heading north and moving at $v_1 = 16~ m/s$ collides in a perfectly inelastic collision with a truck of mass $m_t = 4000~kg$ going East at $v_2 = 9~ m/s$. What is the velocity of the wrecked vehicles just after the collision?

3.	10 pts.	Consider a conical pendulum with a bob of mass $m=60~kg$ on a wire of length $l = 8.0~m$ and making an angle $\theta = 5^\circ$ with the vertical (see the figure). What are the horizontal and vertical components of the force exerted by the wire on the pendulum and what is the radial acceleration of the bob?
4.	10 pts.	A freight car of mass $m_f = 6000~kg$ rolls along rails with negligible friction. The car is brought to rest by a combination of two springs as illustrated in the figure. Both springs obey Hooke's law with $k_1 = 1600~N/m$ and $k_2 = 3400~N/m$. After the first spring compresses a distance $x_1=0.30~m$, the second spring acts with the first to increase the force as additional compression as shown in the graph. The freight car comes to a rest a distance $x_2=0.5~m$ after first contacting the two-spring system. What is the freight car's initial speed? (5,-15)
5.	10 pts.	A catapult launches a rocket at an angle $\theta = 50^\circ$ above the horizontal with an initial speed $v_0 =100~m/s$. The rocket engine immediately starts a burn and for $\Delta t = 2.0~s$ the rocket moves along its initial line of motion with an acceleration $a_0 = 20~m/s^2$. Then its engine fails and the rocket proceeds to move in free-fall. What is the maximum altitude reached by the rocket?

6.

10 pts.

Global warming is a cause for concern because even small changes in the Earth's temperature can have significant consequences. For example, if the Earth's polar ice caps were to melt entirely, the resulting additional water in the oceans would flood many coastal cities. Would it appreciably change the length of a day? Model each polar ice cap as a flat disk of mass $m_i = 1.15\times 10^{19}~kg$, radius $R_i=6\times 10^5~m$, and centered on the axis of rotation of the rest of the Earth. Treat the rest of the Earth as a uniform sphere of mass and radius given in the table of constants. An approximate representation of an Arctic polar cap of these dimensions is shown as the unrealistically red disk in the figure below. In our model, there is a second, Antarctic polar cap that is not visible because of the curve of the Earth. After melting assume the water is evenly spread around the Earth. The additional depth is small compared to the radius of the Earth.

Table of Constants

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	$6.67 \times 10^{-11}~N-m^2/kg^2$	Electron mass	$9.11\times 10^{-31}~kg$
Earth's mass	$5.98 \times 10^{24}~kg$	Earth's radius	$6.37\times 10^6~m$
Earth-Moon distance	$3.84\times 10^8~m$	Earth-Sun distance	$1.496\times 10^{11}~m$

Table of Equations

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

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

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

$$x(t) = \frac{1}{2}at^2 + v_0t + y_0 = \frac{1}{2}(v_0 + v)t ... ...quad v = at + v_0 = \sqrt{2a(x-x_0) + v_0^2} \quad a = g \quad$$

$$a_c = {v^2 \over r} \quad ( \vec v \perp \vec r \quad \vec v... ...ec a = {d \vec p \over dt} \qquad \vec F_{AB} = -\vec F_{BA}$$

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

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

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

$$\Delta\vec p = \int \vec F dt = \langle \vec F \rangle \Delt... ...er dt} \quad \alpha = {a_T \over r} = {d\omega \over dt} \quad$$

$$I = \sum m_i r_i^2 = I_{cm} + MD^2 \quad \vert\vec \tau \v... ...ha \quad L = I \omega \quad L_0 = L_1 \quad v_{cm} = r\omega$$

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

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

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

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

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