Chapter 8: Dynamics II: Motion in a Plane

a. An object of mass m swings in a horizontal circle on a string of length L that tilts downward at angle $u$. Find an expression for the angular velocity$v$.

b. A student ties a $500g$ rock to a$1.0-m$-long string and swings it around her head in a horizontal circle. At what angular speed, in rpm, does the string tilt down at a$10°$ angle?

You’ve taken your neighbor’s young child to the carnival to ride the rides. She wants to ride The Rocket. Eight rocket-shaped cars hang by chains from the outside edge of a large steel disk. A vertical axle through the center of the ride turns the disk, causing the cars to revolve in a circle. You’ve just finished taking physics, so you decide to figure out the speed of the cars while you wait. You estimate that the disk is $5m$ in diameter and the chains are $6m$ long. The ride takes $10s$ to reach full speed, then the cars swing out until the chains are $20°$ from vertical. What is the cars’ speed?

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 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 charged particle of mass m moving with speed v in a plane perpendicular to a magnetic field experiences a force where q is the amount of charge and $F⃗=(qvB,perpendiculartov⃗)$ , B is the magnetic field strength. Because the force is always perpendicular to the particle’s velocity, the particle undergoes uniform circular motion. Find an expression for the period of the motion. Gravity can be neglected.

Two wires are tied to the 2.0 kg sphere shown in FIGURE P8.45. The sphere revolves in a horizontal circle at constant speed.
a. For what speed is the tension the same in both wires?
b. What is the tension?

Two wires are tied to the 300 g sphere shown in FIGURE P8.46. The sphere revolves in a horizontal circle at a constant speed of 7.5 m/s. What is the tension in each of the wires?

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 P8.47 shows that the string traces out the surface of a cone, hence the name.
a. Find an expression for the tension T in the string.
b. Find an expression for the ball’s angular speed v.
c. What are the tension and angular speed (in rpm) for a 500 g ball swinging in a 20-cm-radius circle at the end of a 1.0-m-long string?

The 10 mg bead in FIGURE P8.48 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 ωc the bead sits at the bottom of the spinning loop. When ω > ωc the bead moves out to some angle θ

1. What is ω_c in rpm for the loop shown in the figure?
2. At what value of ω_c in rpm is θ=30^o

In an old-fashioned amusement park ride, passengers stand inside a$5.0m$-diameter hollow steel cylinder with their backs against the wall. The cylinder begins to rotate about a vertical axis. Then the floor on which the passengers are standing suddenly

drops away! If all goes well, the passengers will “stick” to the wall and not slide. Clothing has a static coefficient of friction against steel in the range $0.60$to $1.0$and a kinetic coefficient in the range$0.40$to $0.70.$A sign next to the entrance says “No children under $30kg$allowed.” What is the minimum angular speed, in rpm, for which the ride is safe?