Physics 131-1 Final Exam

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

1. Consider the twins paradox. As the spacefaring 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 spacefaring twin is observing Earth recede from her at a constant velocity. Hence, the spacefaring twin will observe clocks on the Earth to move slowly and the Earthbound twin will age at a slower rate than the spacefaring one. Is this reasoning flawed? How?

2. What would you see in a mirror if you carried it in your hands and ran at (or near) the speed of light?

3. A solid ball and a spherical shell of the same mass and radius are released from rest at the top of an incline. Which one reaches the bottom of the incline first? Why?

4. Consider the one-dimensional position versus time plot below. Does the acceleration of the object change? If so, when is the acceleration largest and smallest? Explain.

5. A ball is tossed in the air. What characteristics of the motion of the ball are constant? What is your evidence?

6. A coin lies on a turntable whose speed is gradually raised from zero. What happens to the magnitude of the friction force on the coin as the speed is increased to a large value?

7. In lab, we showed that for two objects colliding with one another I⃗_{1} = −I⃗_{2} where I⃗ = Δp⃗ is the impulse imparted to each object. Starting from this laboratory observation show that the following vector sum is a constant in time.

∑ p⃗_{n} = p⃗_{1i} + p⃗_{2i} = p⃗_{1f} + p⃗_{2f}

8. How is the work done on an object related to its kinetic energy? How is the work done on an object related to its potential energy? What is your evidence?

9. We found in lab that a significant gravitational attraction still exists even if you are halfway to the moon. Why, then, do astronauts experience weightlessness when they are orbiting a mere 120 km above the Earth?

10. Consider a satellite in a circular orbit around the Earth? What is the net work done on the satellite during one orbit around the Earth? Explain.

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

1.

7 pts.

Galaxy A is reported to be receding from us with a speed of 0.5c. Galaxy B, located in precisely the opposite direction, is found to be receding from us at a velocity of 0.6c.

1. What recessional velocity would an observer on Galaxy A find for our galaxy?

2. What recessional velocity would an observer on Galaxy A find for Galaxy B?

2.

7 pts.

Raindrops fall to Earth from a cloud 1700 m above the ground. If they were not slowed by air resistance, then how fast would they be moving when they struck the Earth? Would it be safe to walk outside during such a rainstorm?

3.

7 pts.

You’ve graduated from college and fulfilled a childhood dream to become a homicide detective in a large city. On your first day at work, you’re investigating the death of a man who was found 4.5 m from the base of his apartment building and 20 m below his balcony. Do you think the death is accidental? Explain.

4.

7 pts.

After a completely inelastic collision, two objects of the same mass and the same initial speed are found to move away together at half their initial speed. Find the angle between the initial and final velocities of the objects.

5.

7 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 θ that the string makes with the axis of the cone that the pendulum sweeps out.

6.

7 pts.

Astronomers have observed that entire galaxies can sometimes collide and combine to form a single large galaxy. Consider the Galaxy Bart shown below on the right. Its shape is essentially a uniform sphere. It has an angular speed about 10^{−17}rad/s, a radius of 10^{21}m, and a mass of 2.0 × 10^{41}kg. Bart was created when the two galaxies Homer and Marge collided as shown in the figure on the left. They each had the shape of a uniform disk. Homer and Marge were nearly identical collections of stars with half the mass of Bart and radii of 7.0 × 10^{20}m. When they collided they became bound at their edges by gravity and started to orbit one another as shown in the figure. Their total linear momentum at the moment of collision was zero. They eventually coalesced into Bart. What was the angular velocity of the two parent galaxies about the point of contact at the moment of the collision? What were their initial speeds before the collision? What is the direction of rotation of Bart?

7.

8 pts.

A communications satellite for a cellular phone company is launched into a circular orbit around the Earth’s equator. To communicate with the satellite it is best if it remains in a fixed position in the sky relative to an observer on the surface of the Earth. This type of orbit is called a geosynchronous one.

1. What should be the height above the Earth’s surface for this satellite to be in a geosynchronous orbit?

2. If the rocket engines misfire and place the satellite in an orbit that is 10^{6} m (1000 km) too low, then how fast does the satellite move across the sky? Is it still usable?

Some Useful Constants

Acceleration of gravity (g) 9.8 m/s^{2}  Speed of light (c)  2.9979 × 10^{8} m/s
Neutron mass  1.68 × 10^{−27} kg Proton mass  1.68 × 10^{−27} kg
Earth mass  5.98 × 10^{24} kg Earth-Sun distance  1.5 × 10^{11} m
Earth radius  6.37 × 10^{6} m  atomic mass unit (u) 1.66 × 10^{−27} kg
1 day  8.64 × 10^{4} s  1 year  3.154 × 10^{7} s
1 hour  3600 s  Sun mass  1.99 × 10^{30} kg
1 mile  1.610 × 10^{3} m Gravitational constant 6.67 × 10^{−11} Nm^{2}/kg^{2}