Circular Motion & Gravitation FR Practice Problems

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1 1) A mass m is attached to a length L of string and hung straight strainght down from a pivot. Small vibrations at the pivot set the mass into circular motion, with the string making an angle θ with the vertical. a) Draw and label all the forces acting on the object. b) Obtain an expression for the speed v of the mass in terms of θ, L, and g.

2 2) The rotor is an amusement park ride that can be modeled as a rotating cylinder, with radius R. A person inside the rotor is held motionless against the sides of the ride as it rotates with a certain velocity. The coefficient of static friction between a person and the sides is µ. a) Derive a formula for the period of rotation T, in terms of R, g, and µ. b) If R = 5 m and µ = 0.5, calculate the value of the period T, in seconds c) Using the answer to part (b), calculate the angular velocity ω, in radians per second.

3 3) An amusement park ride consists of a large cylinder that rotates around its central axis as the passengers stand against the inner wall of the cylinder. Once the passengers are moving at a certain speed v, the floor on which they were standing is lowered, while the passengers remain motionless, pinned to wall of the cylinder as it rotates. a) Draw and label all of the force acting on a passenger of mass m as the cylinder rotates, after the floor has been lowered. b) Describe what conditions must hold to prevent the passengers from sliding down the walls of the container, once the floor has dropped. c) Compare the conditions discussed in part (b) for an adult passenger of mass m and a child passenger of mass m/2.

4 4) A curved road is banked at an angle θ, such that friction is not necessary for a car to stay on the road. A 2500 kg car is traveling at a speed of 25 m/s, and the road has a radius of curvature equal to 40 m. a) Draw a free-body diagram of the situation described above. b) Find angle θ. c) Calculate the magnitude of the force that the road exerts on the car.

5 5) A curved section of highway has a radius of curvature r. The coefficient of friction between standard automobile tires and the surface of the highway is μ s. a) Draw and label all the forces acting on car of mass m traveling along this curved section of highway. b) Compute, in terms of μ s, r,g, and m, the maximum speed with which a car of mass m could make it through the turn without sliding. City engineers are planning on banking this curved section of highway at an angle θ to the horizontal. c) Draw and label all the forces acting on car of mass m traveling along this banked turn. Do not include friction. d) In terms of given variables and fundamental constants, compute the banking angle θ that would allow a vehicle of mass m, driving at speed v, to make it safely through this banked turn under icy (frictionless) conditions.

6 6) A 1,000-kg car makes a turn on a banked curve. The radius of the turn is 300 m, and the turn is inclined at an angle (θ = 30 ). Assume that the turn is frictionless. a) Draw free-body diagram of this situation, and label all the forces acting on the car. b) Calculate the car s maximum speed. c) Calculate the centripetal force on the car.

7 7) A robotic probe lands on a new, uncharted planet. It has determined the diameter of the planet to be 8 x 10 6 m. It weighs a standard 1 kg mass and determines that it weighs only 5 N on this new planet. a) What must the mass of the planet be? b) What is the acceleration due to gravity on this planet? Express your answer in both m/s 2 and g s (where 1 g = 10 m/s 2 ). c) What is the average density of this planet?

8 8) The Earth has a mass of 6 x kg and orbits the Sun in 3.15 x 10 7 s at a constant circular distance of 1.5 x m. a) What is Earth rate of centripetal acceleration around the Sun? b) What is the magnitude of the gravitational force acting between the Sun and the Earth?

9 9) A satellite orbits a planet of uniform density, ρ, mass M, and radius R at a height h above the surface of the planet. The orbital period of the satellite is T. a) Derive an expression for the density of the planet in terms of h, R, G, and T only. b) What is the approximate density if h << R?

10 10) A spacecraft is positioned between the Earth and the Moon such that the gravitational forces on the spacecraft exerted by the Earth and the Moon cancel. a) Is this position closer to the Moon, closer to the Earth, or halfway in between? b) Are the gravitational forces on the spaceship (the force exerted by the Moon, and the force exerted by the Earth) a Newton s 3 rd law force pair?

11 11) A 100 kg body is taken from the surface of the Earth to a satellite in orbit at an altitude of 1.5 R E above the surface of the Earth. a) Determine the weight of the body at the surface of the Earth. b) Determine the weight of the body at an altitude of 1.5 R E above the surface of the Earth. c) What is the fractional change in the weight of the body? d) What is the fractional change in the mass of the body?

12 12) A 100 kg satellite is placed in a low Earth orbit with radius 1.2 x 10 7 m. An identical satellite is to be placed in circular orbit with twice the orbital period. a) Find the speed of the first satellite. b) Find the orbital radius of the second satellite.

13 13) A satellite is in circular orbit around an unknown planet. A second, different satellite also travels in a circular orbit around this planet, but with an orbital radius four times larger than the first satellite. a) Explain what information must be known in order to calculate the speed the first satellite travels in its orbit. b) Compared to the first satellite, how many times faster or slower is the second satellite s speed? c) Bob the bad physics student says: The gravitational force on a satellite in orbit depends inversely on the orbital radius squared. Since the second satellite s orbital radius is four times that of the first satellite, the second satellite experiences one-sixteenth the gravitational force that is exerted on the first satellite. Explain what it wrong with Bob s explanation.

14 14) A space shuttle orbits the Earth 300 km above the equator. a) Explain why it would be impractical for the shuttle to orbit 10 km above the Earth s surface (about 1 km higher than the top of Mount Everest). b) A geosynchronous orbit means that the shuttle will always remain over the same spot on Earth. Explain and describe the calculations you would perform in order to determine whether this orbit is geosynchronous. You should not actually carry out the calculations, just describe them in words and show them in symbols. c) The radius of the Earth is 6,400 km. At the altitude of the space shuttle, what fraction of the surface gravitational field g does the shuttle experience? d) When the shuttle was on Earth before launch, the shuttle s mass (not including any fuel) was 2 x 10 6 kg. At the orbiting altitude, what is the shuttle s mass, not including fuel?

15 15) Find the magnitude of the gravitational field strength g at an point P along the perpendicular bisector between two equal masses, M and M, that are separated by a distance 2b as shown below:

16 16) At a height h above the surface of the Earth, show that if h << R E, the acceleration due to gravity at that point can be approximated by the expression where R E is the radius of the Earth and g is the acceleration due to gravity at the Earth s surface.

17 17) Two stars, each of mass M, form a binary system. The stars orbit about a point a distance R from the center of each star, as shown above. The stars themselves each have a radius r. a) In terms of given variables and fundamental constants, what is the force each star exerts on the other? b) In terms of given variables and fundamental constants, what is the magnitude of the gravitational field at the surface of one of the stars due only to its own mass? c) In terms of given variables and fundamental constants, what is the magnitude of the gravitational field at the midpoint between the two masses? d) Explain why the stars don t crash into each other due to the gravitational force between them.

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