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Chapter 5: Problem 15

At an amusement park there is a ride in which cylindrically shaped chambers spin around a central axis. People sit in seats facing the axis, their backs against the outer wall. At one instant the outer wall moves at a speed of \(3.2 \mathrm{~m} / \mathrm{s},\) and an \(83-\mathrm{kg}\) person feels a \(560-\mathrm{N}\) force pressing against his back. What is the radius of a chamber?

Short Answer

Expert verified
The radius of the chamber is 1.52 meters.

Step by step solution

01

Understand the Problem

The problem involves a spinning cylindrical ride where a person feels a centripetal force due to the rotation. We need to find the radius of the ride based on given force and speed.
02

Identify Known Values

We have the following values: - Speed (\( v \)) = 3.2 m/s - Mass of person (\( m \)) = 83 kg- Centripetal force (\( F_c \)) = 560 N.
03

Use the Centripetal Force Formula

The formula for centripetal force is given by \( F_c = \frac{m v^2}{r} \). We need to solve this equation for the radius (\( r \)).
04

Solve for the Radius

Rearrange the formula to solve for the radius: \[ r = \frac{m v^2}{F_c} \].
05

Substitute the Known Values

Plug in the known values into the rearranged formula:\[ r = \frac{83 \times (3.2)^2}{560} \].
06

Calculate the Result

Calculate the squared speed: \( 3.2^2 = 10.24 \). Now multiply by the mass: \( 83 \times 10.24 = 850.72 \). Divide by the force: \[ r = \frac{850.72}{560} = 1.5192 \text{ m} \].
07

Round Off the Result

Round the result for the radius to three significant figures: \( r = 1.52 \text{ m} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rotational Motion
Rotational motion is a type of motion where an object moves in a circular path around a central point or axis. In the context of the amusement park ride, the chamber rotates around its central axis, causing everything inside to move in circles.
Rotational motion is characterized by three main parameters:
  • Angular velocity: the rate at which the object rotates around the axis. It is usually measured in radians per second.
  • Radius of rotation: the distance from the central axis to the point of interest. In our exercise, it's the distance to the person's back resting on the wall.
  • Centripetal force: a force that acts on any object moving in a circle and is directed toward the center of the circle. It keeps the object moving in its circular path.
The spinning ride you're analyzing exemplifies rotational motion, where the riders experience the effects of rotation directly through the centripetal force they feel pushing them against the wall.
Centripetal Acceleration
Centripetal acceleration occurs when an object moves in a circular path. Despite moving at a constant speed, the direction of the object continuously changes, resulting in acceleration toward the center of the circle.
The formula to calculate centripetal acceleration (\( a_c \)) is given by:\[ a_c = \frac{v^2}{r} \]where \( v \) is the speed of the object, and \( r \) is the radius of the circle.
In the amusement park ride:
  • The speed of the outer wall is given as 3.2 m/s.
  • We worked out the radius to be 1.52 m using the centripetal force formula as given in the original solution.
  • Plugging these values into the centripetal acceleration formula allows us to explore further into the dynamic nature of circular motion.
This acceleration explains why the riders feel like they're being pushed toward the outer wall鈥攊t's not an outward force, but rather the absence of any force in that direction counteracting the centripetal acceleration.
Circular Motion
Circular motion involves any object moving along a circular path. It doesn't have to be in a perfect circle鈥攊t can be an ellipse or any similar shape, but is most simply understood as motion in a circle.
Key concepts of circular motion include:
  • Constant speed: Although the speed remains the same, the direction constantly changes.
  • Velocity: Because direction changes, velocity is a vector quantity and is constantly changing as well.
  • Centripetal force: Needed to maintain circular motion, directed toward the center. Without this force, the object would fly off in a straight line due to inertia.
In the amusement park ride, the seats move along a circular path as the chamber spins, demonstrating perfectly the principles of circular motion. Each aspect of the motion interconnects with the forces and accelerations discussed, creating the thrilling experience of being pushed back against the chamber wall.

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Most popular questions from this chapter

The earth rotates once per day about an axis passing through the north and south poles, an axis that is perpendicular to the plane of the equator. Assuming the earth is a sphere with a radius of \(6.38 \times 10^{6} \mathrm{~m}\), determine the speed and centripetal acceleration of a person situated (a) at the equator and (b) at a latitude of \(30.0^{\circ}\) north of the equator.

The hammer throw is a track-and-field event in which a \(7.3-\mathrm{kg}\) ball (the "hammer") is whirled around in a circle several times and released. It then moves upward on the familiar curving path of projectile motion and eventually returns to earth some distance away. The world record for this distance is \(86.75 \mathrm{~m}\), achieved in 1986 by Yuriy Sedykh. Ignore air resistance and the fact that the ball is released above the ground rather than at ground level. Furthermore, assume that the ball is whirled on a circle that has a radius of \(1.8 \mathrm{~m}\) and that its velocity at the instant of release is directed \(41^{\circ}\) above the horizontal. Find the magnitude of the centripetal force acting on the ball just prior to the moment of release.

At amusement parks, there is a popular ride where the floor of a rotating cylindrical room falls away, leaving the backs of the riders "plastered" against the wall. Suppose the radius of the room is \(3.30 \mathrm{~m}\) and the speed of the wall is \(10.0 \mathrm{~m} / \mathrm{s}\) when the floor falls away. (a) What is the source of the centripetal force acting on the riders? (b) How much centripetal force acts on a \(55.0-\mathrm{kg}\) rider? (c) What is the minimum coefficient of static friction that must exist between a rider's back and the wall, if the rider is to remain in place when the floor drops away?

A curve of radius \(120 \mathrm{~m}\) is banked at an angle of \(18^{\circ} .\) At what speed can it be negotiated under icy conditions where friction is negligible?

A motorcycle has a constant speed of \(25.0 \mathrm{~m} / \mathrm{s}\) as it passes over the top of a hill whose radius of curvature is \(126 \mathrm{~m}\). The mass of the motorcycle and driver is \(342 \mathrm{~kg}\). Find the magnitude of (a) the centripetal force and (b) the normal force that acts on the cycle.
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