Problem 5 - College Physics

Question

The "Giant Swing" at a county fair consists of a vertical central shaft with a number of horizontal arms attached at its upper end. (See Figure 6.28.) Each arm supports a seat suspended from a 5.00-m-long rod, the upper end of which is fastened to the arm at a point \(3.00 \mathrm{~m}\) from the central shaft. (a) Make a free-body diagram of the seat, including the person in it. (b) Find the time of one revolution of the swing if the rod supporting the seat makes an angle of \(30.0^{\circ}\) with the vertical. (c) Does the angle depend on the weight of the passenger for a given rate of revolution?

Step-by-Step Solution

Answer

(a) Free-body diagram with vertical and horizontal tension components. (b) 5.50 m. (c) No, angle does not depend on passenger weight.

Step 1: Analyze the System and Create a Free-Body Diagram

The seat and the person are in circular motion. The forces acting on the person are the gravitational force pointing downward and the tension in the rod. Break the tension force into its vertical and horizontal components. The vertical component of tension balances the gravitational force, while the horizontal component provides the centripetal force for circular motion.

Step 2: Identify Forces and Equations

In the vertical direction, the forces are in equilibrium: the vertical component of the tension ( $t heta$) balances the gravitational force ( $mg$ ). In the horizontal direction, the horizontal component of tension provides centripetal force: $t heta$ = $mω^2r$ .

Step 3: Calculate the Length of the Horizontal Component

Since the rod is \(5.00 \, m\) long and makes an angle of \(30.0^{\circ}\) with the vertical, the horizontal distance from the central shaft is \(r = 3.00 + 5.00 \cdot \sin(30.0^{\circ}) = 3.00 + 5.00 \cdot 0.5 = 5.50 \, m\).

Step 4: Relate Angular Velocity to Tangential Velocity

The vertical component of tension is \(t \, \cos(30.0^{\circ}) = mg\) and the horizontal component is \(t \, \sin(30.0^{\circ}) = m \omega^2 r\). Substitute \(t\) from the vertical equation into the horizontal equation to find \(\omega\) (angular velocity).

Step 5: Calculate the Time for One Revolution

Solve the equation from Step 4 for angular velocity:\[ \omega = \sqrt{\frac{g \cdot \tan(30.0^{\circ})}{r}} \]For one revolution, the time period \(T\) is related to angular velocity\[ T = \frac{2\pi}{\omega} \]Substitute \( r = 5.50 \, m \) and \( g = 9.81 \, m/s^2 \) into the equation for \( \omega \), calculate \( \omega \) first and then \( T \).

Step 6: Analyze the Angle's Dependency on Weight

The angle \(\theta\) is given by \(\tan^{-1}(\frac{g}{\omega^2 r})\), which depends on gravitational acceleration \(g\) and the centripetal force provided by \(\omega^2 r\). Since \(g\) and \(r\) are constants, and \(\omega\) is independent of mass, \(\theta\) does not depend on the weight of the passenger.

Key Concepts

Free-Body Diagram

A free-body diagram is a simple and effective way to represent the forces acting on an object. In circular motion scenarios, like the "Giant Swing" at a county fair, understanding these forces helps us analyze how the system is balanced and how it functions. For the swing seat and the person on it:

• The gravitational force acts downward, pulling both the seat and person towards the Earth's center.
• The tension in the rod is the force that holds the person up and allows them to swing in a circle.

The tension can be split into two components:

• Vertical component: Balances the gravitational force, keeping the system in place vertically.
• Horizontal component: Provides the centripetal force that keeps the seat moving in a circle.

Creating and analyzing a free-body diagram ensures all forces are accounted for, helping us solve further aspects of the problem.

Centripetal Force

Centripetal force is critical for understanding circular motion. It is the inward force required for an object to move in a curve or circle. In the context of our "Giant Swing":"

• This force is provided by the horizontal component of tension in the rod that connects the seat and person to the central shaft.
• Without this force, the seat would fly out in a straight line, rather than follow the circular path.

The equation for centripetal force is:\[ F_c = m \omega^2 r \]where:

• \( m \) is the mass of the object (in this case, the person plus seat).
• \( \omega \) is the angular velocity.
• \( r \) is the distance from the axis of rotation.

Understanding how centripetal force works helps in appreciating why the seat can swing in circles without veering off.

Angular Velocity

Angular velocity is how fast an object spins around a central point. It's like the "speed" of rotation. For the "Giant Swing":

• Angular velocity, \( \omega \), determines how quickly you complete one full circle.
• It connects to centripetal force and the tension in the rod since their relationship governs how tightly or quickly you swing.

The formula relating the time period of one revolution \( T \) and angular velocity is:\[ T = \frac{2\pi}{\omega} \]This formula helps calculate how long it takes to complete one full revolution, crucial for rides like this to ensure safety and predictability during operation.

Equilibrium of Forces

Equilibrium ensures that all forces acting on the swing seat are balanced, so it runs smoothly. In the vertical direction of the "Giant Swing":

• The gravitational force pulling down is balanced by the vertical component of tension in the rod.
• Mathematically represented as \( T \cos(\theta) = mg \).

On the horizontal plane:

• The horizontal tension component equals the centripetal force required to keep the seat moving in a circle.

With these principles:

• Vertical and horizontal forces working together maintain the seat's position and motion.
• The angle of the swing (\(\theta\)) is independent of the passenger's weight since it relies on constant values like gravitational acceleration and angular velocity.

In summary, achieving equilibrium in forces guarantees smooth functioning and eliminates unexpected motion or instability of the swing.