Champlain College Physics 203-NYA

DynamicsCircular motion

List of problems in OpenStax University Physics, Volume 1,chapter 6, section 3:Conceptual questions: 6, 7, 8, 9.Problems: 65, 67, 69, 70, 75, 76, 77, 78, 120, 121.Gravitation and Satellites:chapter 13, sections 1, and 4:Conceptual questions: 6, 7.Problems: 13, 15, 36, 37, 39, 40, 41, 46, 59, 61, 63.

Additional Problems

1. A gymnast in a circus show is swung in a circle by a 12 m cable attached to a verticalrotating pole, as the drawing shows. Suppose the mass of the gymnast is 65 kg.

(a) Determine the tension in the cable.

(b) Find the speed of the gymnast on its circular path.

Solution: The tension is T = 1507 N and the speed of the gymnast is v = 15.1 m/s.

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Champlain College Physics 203-NYA

2. A 5 kg mass is attached to a vertical rotating rod 4 m in length, by two ropes of identicallengths (L = 2.5 m). When the rod rotates with an angular speed of 30 rev/min, themass is held by the ropes in a horizontal circular motion as shown. Find the tensions T1and T2 in each ropes.

Solution: T1 = 92.3 N, T2 = 31.1 N.

3. A 1835 kg car takes an 80 m radius curb along a road inclined 15◦ with respect to thehorizontal with a constant speed of 108 km/h. What must be the minimum frictioncoefficient between the tires and the road, so the car does not slide?

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Champlain College Physics 203-NYA

Solution: µs = 0.673

4. A specific race track has a banking angle of 36◦ in one of its turn with a 50 m radius ofcurvature. How fast can a pilot make this turn if he is driving a Porsche 911 of 1380 kg,equipped with Michelin tires with a coefficient of static friction of 0.5 in normal racingsituations? Give your answer in km/h.

Solution: vmax = 110.6 km/h

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Champlain College Physics 203-NYA

Gravitation and Satellite motion

5. Jupiter’s moon Io has a radius of 1815 km, a mass of 4.7 × 10−5 that of Jupiter, and anorbital radius about Jupiter that is 5.95 times Jupiter’s own radius. Given that Jupiter’smass is 318 times that of the Earth and that its radius is 26 times larger than Earth’s,calculate the acceleration due to gravity on Io at the point nearest Jupiter (A) and thepoint farthest from Jupiter (B). MEarth = 5.976 × 1024 kg, REarth = 6.374 × 106 m.

Solution: aA = 1.68 m/s2, aB = 1.95 m/s2

6. The Little Prince (a character in a book by Antoine de Saint-Exupery) lives on thespherically symmetric asteroid B-612. The density of asteroids, including B-612, is5.2 × 103 kg/m3. Assume that the asteroid does not rotate. The Little Prince noticedthat he felt lighter whenever he walked quickly around his asteroid. In fact, he foundthat he felt weightless and started to orbit the asteroid like a satellite just above thesurface whenever he speeded up to 2 m/s.

Estimate the radius of the asteroid from these data.

Solution: r = 1659 m.

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Champlain College Physics 203-NYA

7. The mass of the planet Mars is 6.42 × 1023 kg, its radius is 3393 km, and a solar day onMars lasts 24h, 39min and 35s.

(a) What would be the weight of a 500 kg robot sent on this planet?

(b) Calculate the period (in hours) of a satellite in circular orbit, 95 km above thesurface of Mars.

(c) At which altitude above the martian surface, must one place a satellite so it followsa synchronous 1 orbit?

Solution:

(a) W = 1860 N

(b) T = 1.74 h

(c) h = 1.7 × 107 m

8. Two identical satellites P and Q are in circular orbits around Earth. The orbital radiusof P is 1.5 times smaller than the one of Q. What is the ratio of their rotation period(TP/TQ)?

Solution: TP/TQ = 0.544

1For Mars we say areosynchronous, and for Earth geosynchronous. Do you know why?

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Champlain College Physics 203-NYA

9. You are working with an international team of scientists trying to send a robot on Titan,the largest of Saturn’s satellites. The mass of Titan is 13.5 × 1022 kg, its radius is 2575 kmand rotates about its axis in 15.945 days. 2

(a) What would be the weight of a 500 kg robot sent on Titan?

(b) At which altitude above Titan’s surface, must your team place a satellite so itfollows a synchronous orbit and remains in constant contact with the robot on thesurface?

(c) The rotation of Titan is itself synchronous, meaning that it takes the same time torotate on itself and around Saturn, thus always facing Saturn the same way 3. IfTitan is observed to have an orbital radius of 1222 × 103 km, Calculate the mass ofSaturn.

Solution:

(a) W = 679 N

(b) h = 7.31 × 107 m

(c) M = 5.69 × 1026 kg

2The rotation period was determined through the comparison of data taken by Voyager 1 (1980) andHubble Space Telescope (1994).

3Our Moon, also has a synchronous orbit, thus always showing the same face.

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