Ch 08: Dynamics II: Motion in a Plane

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 08: Dynamics II: Motion in a Plane

Problem 51b

Chapter 8, Problem 51b

In an amusement park ride called The Roundup, passengers stand inside a 16-m-diameter rotating ring. After the ring has acquired sufficient speed, it tilts into a vertical plane, as shown in FIGURE P8.51. What is the longest rotation period of the wheel that will prevent the riders from falling off at the top?

Verified step by step guidance

Step 1: Begin by identifying the forces acting on the riders at the top of the rotating ring. At this position, the centripetal force must be sufficient to counteract the gravitational force pulling the riders downward.

Step 2: Use the condition for preventing the riders from falling off. The centripetal force is provided by the normal force and gravity. At the minimum speed, the normal force becomes zero, and the centripetal force is entirely provided by gravity. The equation for centripetal force is:

F

_c = mg, where

m

is the mass of the rider and

g

is the acceleration due to gravity.

Step 3: Relate the centripetal force to the rotational motion of the ring. The centripetal force is also given by:

F

_c = mv²/r, where

v

is the tangential speed and

r

is the radius of the ring. Set this equal to the gravitational force:

m g = m v

²/r.

Step 4: Solve for the tangential speed

v

. Cancel out the mass

m

from both sides of the equation:

v

² = gr. Take the square root to find

v = √(g r)

. Here,

r

is half the diameter of the ring, so

r = 16/2 = 8

meters.

Step 5: Relate the tangential speed to the rotation period

T

. The tangential speed is given by

v = 2π r / T

. Substitute

v = √(g r)

into this equation and solve for

T

T = 2π r /√(g r)

. This gives the longest rotation period that prevents the riders from falling off.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Centripetal Force

Centripetal force is the net force acting on an object moving in a circular path, directed towards the center of the circle. In the context of the amusement park ride, this force is necessary to keep the riders in circular motion as the ring rotates. At the top of the ride, the gravitational force and the normal force from the ride must provide the required centripetal force to prevent the riders from falling.

Gravitational Force

Gravitational force is the attractive force between two masses, which in this case is the force acting on the riders due to Earth's gravity. At the top of the ride, this force acts downward and must be balanced with the centripetal force required to keep the riders in circular motion. Understanding the interplay between gravitational force and centripetal force is crucial for determining the conditions under which riders remain safely in their seats.

Period of Rotation

The period of rotation is the time it takes for an object to complete one full revolution around a circular path. In this scenario, the period is related to the speed of the ride and the radius of the circular path. By analyzing the relationship between the period, speed, and radius, one can determine the maximum period that allows the centripetal force to be sufficient to keep the riders from falling off at the top of the ride.

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Related Practice

Textbook Question

A 2.0 kg pendulum bob swings on a 2.0-m-long string. The bob's speed is 1.5 m/s when the string makes a 15° angle with vertical and the bob is moving toward the bottom of the arc. At this instant, what are the magnitudes of the tension in the string?

Textbook Question

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. For what speed is the tension the same in both wires?

Textbook Question

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.

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Textbook Question

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.

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Textbook Question

Suppose you swing a ball of mass m in a vertical circle on a string of length L. As you probably know from experience, there is a minimum angular velocity ω_min you must maintain if you want the ball to complete the full circle without the string going slack at the top. Find an expression for ω_min.

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?