### Educational Content: Spring-Wheel System Dynamics #### Problem Statement: A wheel is attached to a spring with the following characteristics: - **Mass of the Wheel (m):** 20 kg - **Radius of the Wheel (r):** 0.6 m - **Radius of Gyration (k\_g):** 0.4 m - **Spring’s Unstretched Length (L\_0):** 1.0 m - **Stiffness Coefficient of the Spring (k):** 2.0 N/m The system setup involves the wheel being released from rest at **State 1**, where the angle between the spring and the vertical is \( \theta = 30^\circ \). The wheel rolls without slipping and reaches **State 2**, where the angle \( \theta = 0^\circ \). At this point, the spring’s length is \( L_2 = 4 \) m. The objective is to determine the angular velocity of the wheel at **State 2**. #### Diagram Explanation: The diagram shows two states of the wheel-spring system: - **State 1:** - The spring is attached to the wheel. - The angle between the spring and the vertical is \( \theta = 30^\circ \). - The initial rest position, where the wheel begins to roll. - **State 2:** - The angle between the spring and the vertical is \( \theta = 0^\circ \). - The spring is fully stretched to the length of 4 m. The diagram visually represents the wheel at different positions, indicating the transition from the rest position to the fully stretched state along with the spring. #### Question: Determine the angular velocity of the wheel at **State 2** (in radians per second) to two decimal places. --- This problem involves concepts of rotational dynamics and energy conservation, requiring calculations based on the spring’s potential energy, kinetic energy of the wheel, and the conditions of rolling without slipping.
### Educational Content: Spring-Wheel System Dynamics #### Problem Statement: A wheel is attached to a spring with the following characteristics: - **Mass of the Wheel (m):** 20 kg - **Radius of the Wheel (r):** 0.6 m - **Radius of Gyration (k\_g):** 0.4 m - **Spring’s Unstretched Length (L\_0):** 1.0 m - **Stiffness Coefficient of the Spring (k):** 2.0 N/m The system setup involves the wheel being released from rest at **State 1**, where the angle between the spring and the vertical is \( \theta = 30^\circ \). The wheel rolls without slipping and reaches **State 2**, where the angle \( \theta = 0^\circ \). At this point, the spring’s length is \( L_2 = 4 \) m. The objective is to determine the angular velocity of the wheel at **State 2**. #### Diagram Explanation: The diagram shows two states of the wheel-spring system: - **State 1:** - The spring is attached to the wheel. - The angle between the spring and the vertical is \( \theta = 30^\circ \). - The initial rest position, where the wheel begins to roll. - **State 2:** - The angle between the spring and the vertical is \( \theta = 0^\circ \). - The spring is fully stretched to the length of 4 m. The diagram visually represents the wheel at different positions, indicating the transition from the rest position to the fully stretched state along with the spring. #### Question: Determine the angular velocity of the wheel at **State 2** (in radians per second) to two decimal places. --- This problem involves concepts of rotational dynamics and energy conservation, requiring calculations based on the spring’s potential energy, kinetic energy of the wheel, and the conditions of rolling without slipping.
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Question
The wheel is attached to the spring. The mass of the wheel is m=20 kg. The radius of the wheel is 0.6m. The radius of gyration kG=0.4 m. The spring’s unstretched length is L0=1.0 m. The stiffness coefficient of the spring is k=2.0 N/m. The wheel is released from rest at the state 1 when the angle between the spring and the vertical direction is θ=30°. The wheel rolls without slipping and passes the position at the state 2 when the angle is θ=0°. The spring’s length at the state 2 is L2=4 m.
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Transcribed Image Text:### Educational Content: Spring-Wheel System Dynamics
#### Problem Statement:
A wheel is attached to a spring with the following characteristics:
- **Mass of the Wheel (m):** 20 kg
- **Radius of the Wheel (r):** 0.6 m
- **Radius of Gyration (k\_g):** 0.4 m
- **Spring’s Unstretched Length (L\_0):** 1.0 m
- **Stiffness Coefficient of the Spring (k):** 2.0 N/m
The system setup involves the wheel being released from rest at **State 1**, where the angle between the spring and the vertical is \( \theta = 30^\circ \). The wheel rolls without slipping and reaches **State 2**, where the angle \( \theta = 0^\circ \). At this point, the spring’s length is \( L_2 = 4 \) m.
The objective is to determine the angular velocity of the wheel at **State 2**.
#### Diagram Explanation:
The diagram shows two states of the wheel-spring system:
- **State 1:**
- The spring is attached to the wheel.
- The angle between the spring and the vertical is \( \theta = 30^\circ \).
- The initial rest position, where the wheel begins to roll.
- **State 2:**
- The angle between the spring and the vertical is \( \theta = 0^\circ \).
- The spring is fully stretched to the length of 4 m.
The diagram visually represents the wheel at different positions, indicating the transition from the rest position to the fully stretched state along with the spring.
#### Question:
Determine the angular velocity of the wheel at **State 2** (in radians per second) to two decimal places.
---
This problem involves concepts of rotational dynamics and energy conservation, requiring calculations based on the spring’s potential energy, kinetic energy of the wheel, and the conditions of rolling without slipping.
Expert Solution
Introduction:
The total mechanical energy of system is conserved. The wheel rolls down. We are given the radius of gyration of wheel. We can thus find the moment of inertia of wheel. There are 2 states. The change in spring potential energy of the spring is equal to the change in kinetic energy due to energy conservation. Hence the change in potential energy is the sum of rotational and translational kinetic energy of system.
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