8. Centripetal Forces & Gravitation

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?

(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?

(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?

(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?

(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.

(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.

(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?

(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>

(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.

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.

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.]

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?

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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(c) Can the bead ride as high as the center of the circle (θ = 90°)? Explain.

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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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>