8. Centripetal Forces & Gravitation

2.0 kg ball swings in a vertical circle on the end of an 80-cm-long string. The tension in the string is 20 N when its angle from the highest point on the circle is θ = 30°. What is the ball's speed when θ = 30°?

The 10 mg bead in FIGURE CP8.69 is free to slide on a frictionless wire loop. The loop rotates about a vertical axis with angular velocity ω. If ω is less than some critical value ω꜀, the bead sits at the bottom of the spinning loop. When ω > ω꜀, the bead moves out to some angle θ. What is ω꜀ in rpm for the loop shown in the figure?

The 'Giant Swing' at a county fair consists of a vertical central shaft with a number of horizontal arms attached at its upper end. Each arm supports a seat suspended from a cable $5.00$ m long, and the upper end of the cable is fastened to the arm at a point $3.00$ m from the central shaft (Fig. E$5.50$). Find the time of one revolution of the swing if the cable supporting a seat makes an angle of $30.0°$ with the vertical.

A 4.4-cm-diameter, 24 g plastic ball is attached to a 1.2-m-long string and swung in a vertical circle. The ball's speed is 6.1 m/s at the point where it is moving straight up. What is the magnitude of the net force on the ball? Air resistance is not negligible.

A small remote-controlled car with mass $1.60$ kg moves at a constant speed of $v=12.0$ m/s in a track formed by a vertical circle inside a hollow metal cylinder that has a radius of $5.00$ m (Fig. E$5.45$). What is the magnitude of the normal force exerted on the car by the walls of the cylinder at point $A$ (bottom of the track)?

In another version of the 'Giant Swing' (see Exercise $5.50$), the seat is connected to two cables, one of which is horizontal (Fig. E$5.51$). The seat swings in a horizontal circle at a rate of $28.0$ rpm (rev/min). If the seat weighs $255$ N and an $825$-N person is sitting in it, find the tension in each cable.

(II) At what rate must a cylindrical spaceship rotate (Fig. 6–32) if occupants are to experience simulated gravity of 0.70 g? Assume the spaceship’s diameter is 28 m, and give your answer as the time needed for one revolution.

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A small 4kg block is tied to the end of 3m string and slides around in a circle on a frictionless table. Suppose the string will break if the tension exceeds 50N. Find the maximum speed the block can have without breaking the string.

A 30 g ball rolls around a 40-cm-diameter L-shaped track, shown in FIGURE P8.53, at 60 rpm. What is the magnitude of the net force that the track exerts on the ball? Rolling friction can be neglected. Hint: The track exerts more than one force on the ball.

Which of the following methods will correctly set the desired centripetal force for an object moving in a horizontal circle of radius $r$ at constant speed $v$ and mass $m$?

A conical pendulum is formed by attaching a ball of mass m to a string of length L, then allowing the ball to move in a horizontal circle of radius r. FIGURE P8.48 shows that the string traces out the surface of a cone, hence the name. Find an expression for the ball's angular speed ω.

A 5.0 g coin is placed 15 cm from the center of a turntable. The coin has static and kinetic coefficients of friction with the turntable surface of μ_s = 0.80 and μ_k = 0.50. The turntable very slowly speeds up to 60 rpm. Does the coin slide off?

A small remote-controlled car with mass $1.60$ kg moves at a constant speed of $v = 12.0$ m/s in a track formed by a vertical circle inside a hollow metal cylinder that has a radius of $5.00$ m (Fig. E$5.45$). What is the magnitude of the normal force exerted on the car by the walls of the cylinder at point $B$ (top of the track)?