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Transcribed Image Text:### Swinging Motion of a Square Object Suspended by a Hook
**Problem Statement:**
A square object of mass \( m \) is constructed of four identical uniform thin sticks, each of length \( L \), attached together. This object is hung on a hook at its upper corner. If it rotates slightly to the left and then released, at what frequency will it swing back and forth?
**Diagram Explanation:**
The diagram shows a square object hanging from a hook at its upper corner. The square is constructed from four identical sticks, each labeled with a length \( L \). The object is oriented in such a way that two sides form a 45-degree angle with the vertical.
**Detailed Diagram Explanation:**
1. **Hook:** Indicates the point of suspension where the square's corner is connected.
2. **Square Sticks:** Four identical sticks of length \( L \) form the square shape.
3. **Length Labels:** Each side of the square is labeled \( L \).
### Solution Approach:
To determine the frequency of the oscillations, we need to calculate the moment of inertia of the square about the point of suspension and then use the principles of rotational motion to find the frequency.
This involves:
1. Calculating the moment of inertia of the square about the pivot point.
2. Using the torque due to gravity to set up the equation of motion.
3. Solving the differential equation to find the angular frequency.
Let’s proceed:
1. **Moment of Inertia Calculation:**
The square can be considered as being composed of four sticks. The moment of inertia of each stick about its end is \( \frac{1}{3}mL^2 \). Using the parallel axis theorem and summing the contributions from all sticks, we find the total moment of inertia about the pivot point.
2. **Setting Up the Equation of Motion:**
The restoring torque caused by gravity will be proportional to the angular displacement. For small angles, the equation resembles that of a simple harmonic oscillator.
3. **Finding the Frequency:**
Solving the simple harmonic oscillator equation provides the frequency of oscillation.
By working through these steps mathematically, students can gain insight into the interplay between rotational dynamics and oscillatory motion. This problem is a practical example of applying physical principles to a real-world scenario.
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