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Chapter 5: Problem 82

In an amusement park rocket ride, cars are suspended from 4.25 -m cables attached to rotating arms at a distance of \(6.00 \mathrm{m}\) from the axis of rotation. The cables swing out at an angle of \(45.0^{\circ}\) when the ride is operating. What is the angular speed of rotation?

Short Answer

Expert verified
The angular speed of the ride is approximately 1.04 rad/s.

Step by step solution

01

Understand the Problem

This problem involves a rotating ride where cars are suspended from cables and swing out at a certain angle. We are asked to find the angular speed of the rotation when the angle is given.
02

Define Given Values

We have the cable length as 4.25 meters, the distance from the axis of rotation to the point where the cable is attached as 6.00 meters, and the angle the cable makes with the vertical as 45.0 degrees.
03

Determine the Geometry

When the cable swings out at 45 degrees, a right triangle is formed. The hypotenuse is the cable length (4.25 m), one leg is the horizontal distance from the vertical to where the cable attaches, and the other leg is part of the rotation radius (which includes the 6.00 meters).
04

Calculate the Radius of Rotation

Calculate the horizontal component of the cable using cosine: \ \[\text{Horizontal component} = 4.25 \times \sin(45^\circ) = 4.25 \times \frac{\sqrt{2}}{2} = 3.01 \ \text{m}\] \ The actual radius of rotation is this horizontal component plus 6.00 m: \ \[R = 6.00 + 3.01 = 9.01 \ \text{m}\]
05

Apply Dynamic Equilibrium in Circular Motion

The system is in dynamic equilibrium, so the tension provides the centripetal force. The component of tension provides the force for circular motion: \ \[ m g \tan(45^\circ) = m \omega^2 R \] \ where \(\omega\) is the angular speed and \(R\) is the total radius we calculated.
06

Solve for Angular Speed

Since \(\tan(45^\circ) = 1\), it simplifies to: \ \[ g = \omega^2 R \] \ So the angular speed \(\omega\) is: \ \[ \omega = \sqrt{\frac{g}{R}} = \sqrt{\frac{9.81}{9.01}} \approx 1.04 \ \text{rad/s} \]
07

Conclusion

The angular speed of the ride when the cars swing out at a 45-degree angle is approximately \(1.04 \ \text{rad/s}\).

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Key Concepts

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

Angular Speed
In circular motion, angular speed plays a crucial role in understanding how fast an object rotates around a central point. Angular speed, often denoted as \(\omega\), indicates how many radians an object travels per second.
A radian is a measure of an angle based on the radius of a circle. One full rotation around a circle is \(2\pi\) radians.
  • The formula to calculate angular speed is \(\omega = \frac{\theta}{t}\), where \(\theta\) is the angle in radians traversed, and \(t\) is the time taken.
  • In our amusement park scenario, we found that \(\omega = \sqrt{\frac{g}{R}} = \sqrt{\frac{9.81}{9.01}} \approx 1.04\; \text{rad/s}\).
This means each car on the ride completes about 1.04 radians per second during its rotation.
Centripetal Force
Centripetal force is vital for keeping an object in circular motion, constantly pulling the object towards the center of its path. Without it, the object would move in a straight line due to inertia.
  • The formula for centripetal force is \(F_c = m \cdot \omega^2 \cdot R\), where \(m\) is mass, \(\omega\) is angular speed, and \(R\) is the radius of rotation.
  • In the dynamical context of our amusement park ride, the tension in the cables acts as the centripetal force, balancing with the gravitational force component for circular motion.
Centripetal force ensures that ride cars maintain their circular paths, providing a thrilling experience for riders.
Trigonometry
Trigonometry is central to solving problems involving angles and lengths, particularly in circular motion scenarios like our amusement park ride.
  • When the cables swing out at a \(45^\circ\) angle, it forms a right triangle with the vertical.
  • Trigonometric functions like sine and cosine help calculate components of vectors, such as calculating the horizontal component of the cable length using \(\sin(45^\circ) = \frac{\sqrt{2}}{2}\).
These trigonometric calculations allow us to determine the radius of rotation crucial for further calculations.
Rotational Dynamics
Rotational dynamics is the branch of physics focusing on the rotational motion of bodies. It considers factors like torque, angular velocity, and angular acceleration to describe and predict the behavior of rotating systems.
  • Key equations in rotational dynamics often parallel linear dynamics but in angular terms, involving aspects such as torque (the rotational equivalent of force).
  • Our exercise incorporates dynamic equilibrium, a situation where net forces are zero, allowing us to effectively solve for angular speed using given forces and moments.
Understanding rotational dynamics helps illuminate how various forces and motions interplay in scenarios like amusement rides, ensuring they are both safe and exhilarating.

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Most popular questions from this chapter

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