toy car is released from rest at a height  above the bottom of the loop of radius , goes through the loop, and then travels along a flat path. If the car just barely passes the top of the loop, find the minimum height  the car must be released from. Neglect energy loss due to friction.

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Chapter1: Units, Trigonometry. And Vectors
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A toy car is released from rest at a height  above the bottom of the loop of radius , goes through the loop, and then travels along a flat path. If the car just barely passes the top of the loop, find the minimum height  the car must be released from. Neglect energy loss due to friction.

The image illustrates a dynamic system involving a car on a track with a loop.

1. **Track Description**:
   - The track starts with a curved ramp, labeled with an initial height \( y_i \).
   - After descending the ramp, the track leads into a vertical loop with radius \( R \).
   - The x-axis serves as the horizontal reference line, and the y-axis as the vertical reference.

2. **Diagram Elements**:
   - The initial position of the car is shown at the top of the ramp at height \( y_i \).
   - As the car moves down the ramp, it gains speed and then enters the loop-the-loop.
   - The car is depicted again at the topmost point of the loop.

3. **Conceptual Focus**:
   - The diagram is likely intended to demonstrate principles of conservation of energy, including potential and kinetic energy transformations.
   - The emphasis can be on explaining how gravitational potential energy at height \( y_i \) is converted into kinetic energy, allowing the car to complete the loop, overcoming the gravitational pull at the top of the loop.
Transcribed Image Text:The image illustrates a dynamic system involving a car on a track with a loop. 1. **Track Description**: - The track starts with a curved ramp, labeled with an initial height \( y_i \). - After descending the ramp, the track leads into a vertical loop with radius \( R \). - The x-axis serves as the horizontal reference line, and the y-axis as the vertical reference. 2. **Diagram Elements**: - The initial position of the car is shown at the top of the ramp at height \( y_i \). - As the car moves down the ramp, it gains speed and then enters the loop-the-loop. - The car is depicted again at the topmost point of the loop. 3. **Conceptual Focus**: - The diagram is likely intended to demonstrate principles of conservation of energy, including potential and kinetic energy transformations. - The emphasis can be on explaining how gravitational potential energy at height \( y_i \) is converted into kinetic energy, allowing the car to complete the loop, overcoming the gravitational pull at the top of the loop.
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