A bead of mass m , with a hole through its center, slides without friction along a vertical hoop of radius R. moves under the combined influence of gravity and a spring attached to the bottom of the hoop. For simplicity, we assume that the equilibrium length of the spring is zero, so that the force due to the spring is -kr, where r the instantaneous length of the spring, as shown. The bead hoop R nail (a) What is the intial energy of the system when the mass is at rest at the top of the hoop? (b) The bead is released at the top of the hoop with negligible speed. How fast is it moving when it reaches the bottom of the hoop? (c) Why would this problem be difficult to solve using a force approach?
A bead of mass m , with a hole through its center, slides without friction along a vertical hoop of radius R. moves under the combined influence of gravity and a spring attached to the bottom of the hoop. For simplicity, we assume that the equilibrium length of the spring is zero, so that the force due to the spring is -kr, where r the instantaneous length of the spring, as shown. The bead hoop R nail (a) What is the intial energy of the system when the mass is at rest at the top of the hoop? (b) The bead is released at the top of the hoop with negligible speed. How fast is it moving when it reaches the bottom of the hoop? (c) Why would this problem be difficult to solve using a force approach?
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Transcribed Image Text:### Problem Statement
A bead of mass \( m \), with a hole through its center, slides without friction along a vertical hoop of radius \( R \). The bead moves under the combined influence of gravity and a spring attached to the bottom of the hoop. For simplicity, we assume that the equilibrium length of the spring is zero, so that the force due to the spring is \(-kr\), where \( r \) is the instantaneous length of the spring, as shown.
#### Questions:
(a) What is the initial energy of the system when the mass is at rest at the top of the hoop?
(b) The bead is released at the top of the hoop with negligible speed. How fast is it moving when it reaches the bottom of the hoop?
(c) Why would this problem be difficult to solve using a force approach?
### Diagram Explanation
The diagram depicts a vertical hoop with a radius \( R \), with a bead located at the top of the hoop. A spring with spring constant \( k \) is attached to the bottom of the hoop and extends upward to connect with the bead. The spring is compressed or stretched as the bead moves. Gravity acts downward on the bead.
- The hoop is a full circle with radius \( R \).
- The spring's length changes dynamically, influencing the motion of the bead.
- A nail is depicted at the bottom center of the hoop from which the spring is attached.
### Considerations for Problem Solving
- **Energy Conservation Approach**: Analyzing the system using potential and kinetic energy may simplify the problem since forces vary continuously with the bead's position.
- **Spring Mechanics**: The force exerted by the spring is proportional to its change in length, complicating direct force analysis.
- **Gravitational Influence**: The change in gravitational potential energy must be considered as the bead moves from top to bottom.
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