Physics 1100 In-Class Problems: Uniform Circular Motion and Gravity

1. In the diagram below, an object travels over a hill, down a valley, and around a loop-the-loop at constant speed v. At each of the specified points draw a free body diagram indicating the directions of the normal force and of the weight. Also indicate the magnitude and direction of the centripetal acceleration, if any. Write down the equations that Newton's Second Law gives you for the points with centripetal acceleration.
   

2. A 1.50-kg rock is being twirled in a circle on a frictionless surface using a horizontal rope. The radius of the circle is 2.00 m and the rope make 100 revolutions in 1.00 minutes. What is the tension in the rope?

3. The rope in question 2 will break when the tension exceeds 1000 N. What will be the speed of the rock just as the rope breaks?

4. Bicycle racetracks, or velodromes, are banked at the ends. If we ignore friction, and the banking angle is θ =25°, what will the maximum speed of a bicycle if it is to move around the end of the track at constant radius r? If the bicyclist goes faster that this value what will happen? If the bicyclist goes slower?
   

5. Suppose we do not ignore friction in question 4 and that the coefficient of static friction is μ_s. How fast can the bicyclist travel and still remain at radius r? How slow could the bicyclist ride and still travel in a circle?

6. The moon circles the earth once every 27.3 days. We have already determined that the mass of the earth is 5.98 × 10²⁴ kg. What is the distance from the center of the earth to the centre of the moon?

7. The earth is a satellite of the Sun. The distance from the sun to the earth is 1.50 × 10¹¹ m. What is the mass of the Sun?
   

8. What is the apparent weight of a 75.0-kg person travelling at 100 km/h (a) over the peak of a hill with radius of curvature equal to 500 m, and (b) at the bottom of a hollow of the same radius?

9. What is the minimum speed for a rollercoaster to be remain in contact with the tracks if it is doing an upside down loop of radius 350 m.

   

10. The brightest four moons of Jupiter were discovered by Galileo with one of his earliest telescopes. These moons, Io, Europa, Ganymede, and Callisto, are called the Galilean moons in his honour. Some of the available data about these moons are given below.
    

    MOON	r (km)	v	T (earthyears)
    Io	4.219 × 105	-	0.004837
    Europa	6.712 × 105	-	-
    Ganymede	-	-	0.0195884
    Callisto	1.853 × 106	-	-

    The radii are from the centre of Jupiter to the centre of the moon in question. One earth year has 365 days. From the above data, determine (a) the mass of Jupiter, (b) the period of Europa, (c) the distance between Jupiter and Ganymede, and (d) the speed of Callisto.

11. The mass of the planet Mercury is 3.30 × 10²³ kg and its radius is 2.439 × 10⁶ m. What would a 65.0-kg person weigh on Mercury? What is the acceleration due to gravity on Mercury.
    

12. Ted and Alice are mutually attracted to one another in the gravitational sense. If Ted's mass is 80.0 kg and Alice's is 55.0 kg and they are 0.150 m apart, what is the magnitude of the attraction on each? Treat both people as spheres.

13. What is the weight of an 80.0-kg cosmonaut on the space station Mir, 420 km above the surface of the earth? What is the acceleration due to gravity?

14. Given that the mass of the moon is 7.35 × 10²² kg, that the distance between the centres of the earth and the moon is 3.85 × 10⁸ m, and that the radius of the earth is 6378 km, find the gravitational pull of the moon on a 75-kg person when the moon is directly overhead. Compare (i.e. take the ratio) this to the person's weight.

15. The distance between the centres of the earth and the moon is 3.85 × 10⁸ m. The moon has a mass which is only 1.29% that of earth. Where would a satellite have to be placed to feel no net gravitational pull from the earth and the moon?

Questions? mike.coombes@kwantlen.ca