Circular Motion Study Guide

Discussion on forces acting perpendicular or parallel to an object's displacement.
Key Principles:
- Forces acting parallel to motion affect speed (acceleration).
- Forces acting perpendicular to motion affect direction.

Scenario: Driving a car forward.
- Possible actions to change motion:
- Speeding up: Pressing the gas leads to static friction force accelerating the car.
- Slowing down: Pressing the brakes leads to kinetic friction slowing down the car.
- Turning left or right: Static friction facilitates the turn to avoid skidding.
Conclusion: Forces are responsible for changes in motion; net force causes this change.

Examining the combined action of parallel and perpendicular forces in motion:
- Forward Acceleration: Forces parallel to motion (e.g., driving forward).
- Direction Change: Forces perpendicular to motion (e.g., turning).
- Result: The car can speed up and turn simultaneously.

Tangential Acceleration (a_t):
- Defined as the rate of change of speed along a path;
- Affects speeding up or slowing down an object.
Centripetal Acceleration (a_c):
- Always directed toward the center of the circular path;
- Defined as acceleration required for circular motion.
- Equation: {a_c = rac{v^2}{r}} where:
- v = velocity of the object,
- r = radius of the circular path.
Net Acceleration:
- Resultant of tangential and centripetal accelerations.
- Can occur in various combinations:
- Only tangential acceleration when moving in a straight line and accelerating.
- Only centripetal acceleration when moving at a constant speed while turning.
- Both when simultaneously turning and speeding up/slowing down.

Setup: Block attached to a string and pushed to move in a circle.
Observation Points: Two key points at approximate positions 2 o'clock (upper right) and 8 o'clock (lower left).
- 2 o'clock Position:
- Velocity: Tangent to the path, directed up to the left.
- Tangential Acceleration: Kinetic friction acting backwards (slowing down).
- Centripetal Acceleration: Tension force pointing towards the center.
- Net Acceleration: Intermediate vector between backward (tangential) and inward (centripetal).
- 8 o'clock Position:
- Velocity: Tangent to the path, directed down to the right.
- Tangential Acceleration: Kinetic friction opposing motion, directed backward.
- Centripetal Acceleration: Tension directed towards the center.
- Net Acceleration: Similar to the 2 o'clock position.

Perpendicular Direction:
- Net force (tension) directed towards aligning with centripetal acceleration:
- T = m a_c (where T = tension, {m} = mass of the block).
Parallel Direction:
- Net force (kinetic friction) affects tangential acceleration:
- f{k} = m at (where f{k} = kinetic friction and {at} is tangential acceleration).

Marble follows a path that dips and rises.
Analysis at Key Points:
- Bottom of the Hill:
- Follows circular motion; normal force upwards exceeds gravitational force downwards:
  - Net Force Equation: FN - mg = mac
  - Results in normal force: F_N = rac{mv^2}{r} + mg
- Top of the Hill:
- Gravitational force downwards exceeds normal force upwards:
  - Net Force Equation: mg - FN = mac
  - Results in normal force: F_N = mg - rac{mv^2}{r}
Comparison: Normal force is greater at the bottom of the hill compared to the top.

Forces parallel to velocity lead to tangential acceleration.
Forces perpendicular to velocity induce centripetal acceleration.
Expressions:
- ext{Net Force Parallel} = m a_t
- ext{Net Force Perpendicular} = m a_c
Clarification: There is no distinct force called centripetal force; it is the net force creating centripetal acceleration, which can consist of multiple forces acting together.