8. Centripetal Forces & Gravitation

Uniform Circular Motion

8. Centripetal Forces & Gravitation

Uniform Circular Motion

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A ball travels on a frictionless circular track at 3m/s. The ball cannot have an acceleration greater than 1.5m/s² or it will go off the track. What is the smallest radius the circular track can have so that the ball stays on the track?

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Textbook Question

At its Ames Research Center, NASA uses its large '20-G' centrifuge to test the effects of very large accelerations ('hypergravity') on test pilots and astronauts. In this device, an arm 8.84 m long rotates about one end in a horizontal plane, and an astronaut is strapped in at the other end. Suppose that he is aligned along the centrifuge's arm with his head at the outermost end. The maximum sustained acceleration to which humans are subjected in this device is typically 12.5g. (b) What is the difference between the acceleration of his head and feet if the astronaut is 2.00 m tall?

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Textbook Question

At its Ames Research Center, NASA uses its large '20-G' centrifuge to test the effects of very large accelerations ('hypergravity') on test pilots and astronauts. In this device, an arm 8.84 m long rotates about one end in a horizontal plane, and an astronaut is strapped in at the other end. Suppose that he is aligned along the centrifuge's arm with his head at the outermost end. The maximum sustained acceleration to which humans are subjected in this device is typically 12.5g. (a) How fast must the astronaut's head be moving to experience this maximum acceleration?

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Textbook Question

A model of a helicopter rotor has four blades, each 3.40 m long from the central shaft to the blade tip. The model is rotated in a wind tunnel at 550 rev/min. (b) What is the radial acceleration of the blade tip expressed as a multiple of g?

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Textbook Question

A model of a helicopter rotor has four blades, each 3.40 m long from the central shaft to the blade tip. The model is rotated in a wind tunnel at 550 rev/min. (a) What is the linear speed of the blade tip, in m/s?

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Textbook Question

The earth has a radius of 6380 km and turns around once on its axis in 24 h. (b) If arad at the equator is greater than g, objects will fly off the earth's surface and into space. (We will see the reason for this in Chapter 5.) What would the period of the earth's rotation have to be for this to occur?

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Textbook Question

The earth has a radius of 6380 km and turns around once on its axis in 24 h. (a) What is the radial acceleration of an object at the earth's equator? Give your answer in m/s2 and as a fraction of g.

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Textbook Question

Communications satellites are placed in a circular orbit where they stay directly over a fixed point on the equator as the earth rotates. These are called geosynchronous orbits. The radius of the earth is 6.37 x 10⁶ m, and the altitude of a geosynchronous orbit is 3.58 x 10⁷ m (≈ 22,000 miles). What are (a) the speed and (b) the magnitude of the acceleration of a satellite in a geosynchronous orbit?

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Textbook Question

The radius of the earth's very nearly circular orbit around the sun is 1.5 x 10^11m. Find the magnitude of the earth's (b) angular velocity

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Textbook Question

The radius of the earth's very nearly circular orbit around the sun is 1.5 x 10^11m. Find the magnitude of the earth's (a) velocity

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Textbook Question

As the earth rotates, what is the speed of (a) a physics student in Miami, Florida, at latitude 26 degrees

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Textbook Question

The earth's radius is about 4000 miles. Kampala, the capital of Uganda, and Singapore are both nearly on the equator. The distance between them is 5000 miles. The flight from Kampala to Singapore takes 9.0 hours. What is the plane's angular velocity with respect to the earth's surface? Give your answer in degrees/h.

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Textbook Question

Peregrine falcons are known for their maneuvering ability. In a tight circular turn, a falcon can attain a centripetal acceleration 1.5 times the free-fall acceleration. What is the radius of the turn if the falcon is flying at 25 m/s?

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Textbook Question

(I) A jet plane traveling 1890 km/h (525 m/s) pulls out of a dive by moving in an arc of radius 4.80 km. What is the plane's acceleration in g's?

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Textbook Question

(II) A bug walks outward from the center of a turntable that is rotating at 65 rpm. If the coefficient of static friction between the bug and the turntable is 0.42, how far out from the center does the bug get before it starts to slip on the turntable? What happens to the bug after it moves farther out past that point?

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Textbook Question

(II) A coin is placed 10.5 cm from the axis of a rotating turntable of variable speed. When the speed of the turntable is slowly increased, the coin remains fixed on the turntable until a rate of 38.0 rpm (revolutions per minute) is reached, at which point the coin slides off. What is the coefficient of static friction between the coin and the turntable?

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Textbook Question

A 200 g block on a 50-cm-long string swings in a circle on a horizontal, frictionless table at 75 rpm. (a) What is the speed of the block?

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Textbook Question

FIGURE P8.54 shows two small 1.0 kg masses connected by massless but rigid 1.0-m-long rods. What is the tension in the rod that connects to the pivot if the masses rotate at 30 rpm in a horizontal circle?

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Textbook Question

Suppose the moon were held in its orbit not by gravity but by a massless cable attached to the center of the earth. What would be the tension in the cable? Use the table of astronomical data inside the back cover of the book.

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Textbook Question

A 200 g block on a 50-cm-long string swings in a circle on a horizontal, frictionless table at 75 rpm. (b) What is the tension in the string?

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Textbook Question

(III) The position of a particle moving in the xy plane is given by r→ = (2.0m) cos [(3.0 rad/s)t ] i hut +(2.0m) sin [(3.0 rad/s)t ] j hut, where r is in meters and t is in seconds.

(b) Calculate the velocity and acceleration vectors as functions of time.

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Textbook Question

(III) The position of a particle moving in the xy plane is given by r→ = (2.0m) cos [(3.0 rad/s)t ] i hut +(2.0m) sin [(3.0 rad/s)t ] j hut , where r is in meters and t is in seconds.

(e) Show that the acceleration vector always points toward the center of the circle.

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Textbook Question

(II) How many revolutions per minute would a 22-m-diameter Ferris wheel need to make for the passengers to feel “weightless” at the topmost point?

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Textbook Question

(II) While fishing, you get bored and start to swing a sinker weight around in a circle below you on a 0.45-m piece of fishing line. The weight makes a complete circle every 0.65 s. What is the angle that the fishing line makes with the vertical? [Hint: See Fig. 5–20.] <IMAGE>

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Textbook Question

(II) Tarzan plans to cross a gorge by swinging in an arc from a hanging vine (Fig. 5–50). If his arms are capable of exerting a force of 1350 N on the vine, what is the maximum speed he can tolerate at the lowest point of his swing? His mass is 78 kg and the vine is 4.8 m long.

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Textbook Question

A small bead of mass m is constrained to slide without friction inside a circular vertical hoop of radius r which rotates about a vertical axis (Fig. 5–58) at a frequency f.

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(a) Determine the angle θ where the bead will be in equilibrium within the hoop—that is, where it will have no tendency to move up or down along the hoop.

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Textbook Question

A train traveling at a constant speed rounds a curve of radius 215 m. A lamp suspended from the ceiling swings out to an angle of 18.5° throughout the curve. What is the speed of the train? [Hint: See Example 4–15.]

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Textbook Question

A small bead of mass m is constrained to slide without friction inside a circular vertical hoop of radius r which rotates about a vertical axis (Fig. 5–58) at a frequency f. <IMAGE>

(b) If ƒ = 2.00 rev/s and r = 25.0cm , what is θ?

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Textbook Question

A small bead of mass m is constrained to slide without friction inside a circular vertical hoop of radius r which rotates about a vertical axis (Fig. 5–58) at a frequency f.

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Textbook Question

In a “Rotor-ride” at a carnival, people rotate in a vertical cylindrically walled “room.” See Fig. 5–52. If the room radius is 5.5 m, and the rotation frequency 0.50 revolutions per second when the floor drops out, what minimum coefficient of static friction keeps the people from slipping down? People on this ride said they were “pressed against the wall.” Is there really an outward force pressing them against the wall? If so, what is its source? If not, what is the proper description of their situation (besides nausea)? [Hint: Draw a free-body diagram for a person.]

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Textbook Question

A flat puck (mass M) is revolved in a circle on a frictionless air hockey table top, and is held in this orbit by a light cord which is connected to a dangling mass (mass m) through a central hole as shown in Fig. 5–51. Show that the speed of the puck is given by v = √mgR/M. <IMAGE>

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8. Centripetal Forces & Gravitation

Uniform Circular Motion

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