Uniform Circular Motion Review
Key Concepts
- Uniform Circular Motion: Motion of objects moving in a circle at constant speed.
- Velocity Vector: Always tangent to the circle, changing direction as object moves.
- Centripetal Acceleration: Always directed towards the center of the circle.
Formulas
-
Centripetal Acceleration ((a_c)):
- (a_c = \frac{v^2}{r})
- If velocity doubles, centripetal acceleration increases by factor of four.
-
Centripetal Force ((F_c)):
- Derived from Newton's Second Law: (F = ma)
- (F_c = m \cdot a_c = \frac{m \cdot v^2}{r})
-
Velocity:
- (v = \frac{2 \pi r}{T})
- Period (T): Time for one complete revolution.
- Frequency (f): (f = \frac{1}{T}), measured in hertz (Hz).
-
Alternative Formula for Centripetal Acceleration:
- (a_c = \frac{4 \pi^2 r}{T^2})
Calculating Tension in Circular Motion
-
Vertical Circle:
- At top/bottom points (A and C), tension: (T = \frac{m v^2}{r})
- Bottom (Point D): (T = F_c + mg)
- Top (Point B): (T = F_c - mg)
-
Horizontal Circle:
- Fast motion: (T \approx F_c)
- With angle: (T = \sqrt{T_x^2 + T_y^2})
- (T_x = \frac{m v^2}{r}), (T_y = mg)
- (\tan(\theta) = \frac{T_y}{T_x})
Application to Hills and Valleys
-
Bottom of Valley:
- Normal force: (N = F_c + mg)
-
Top of Hill:
- Normal force: (N = mg - F_c)
- If (mg < F_c), object may lose contact with ground.
Additional Resources
- Example problems and further explanations can be found in linked videos, particularly for calculating normal force on hills.
For more detailed examples and calculations, access supplementary videos provided in the lecture resource links.