P210A+CH+6
Chapter 6: Introduction to Uniform Circular Motion and Gravitation
Page 1: Course Information
Course Title: PHYS 210A General Physics
Instructor: Professor Sara Kassis
Department: Physics and Astronomy
Contact: Sara.kassis@sonoma.edu
Page 2: Key Definitions
Uniform Circular Motion:
Movement in a circular path at a constant speed.
Rotational Motion:
Points in an object move in circular paths around a single point.
Translational Motion:
Motion without rotation.
Example:
A rotating hockey puck moving across ice combines both rotational and translational motion.
Page 3: Rotation Angle and Radians
Rotation Angle (Δq):
Ratio of arc length (Δs) to radius of curvature (r): Δq = Δs / r
One revolution equals 360 degrees or 2π radians.
1 full circle = 2π radians; radians are used for calculations.
Page 4: Velocity Definitions
Angular Velocity (ω):
Rate of change of angle (q), units: rad/s.
Formula: ω = Δq / Δt
Linear Velocity (v):
Rate of change of arc length (s), units: m/s.
Relation to angular velocity: v = rω, also expressed as v = 2πr / T, with T as the period of one revolution.
Direction:
Angular velocity: clockwise or counterclockwise.
Linear velocity: tangent to the circular path.
Page 5: Centripetal Acceleration
Centripetal Acceleration (ac):
Acceleration of an object moving in a circular path, directed towards the center of the circle.
Relations: ac = v² / r
Characteristics:
Increases with higher speeds and smaller radius curves.
Page 6: Centrifuge Example
Centrifuge:
Simulates gravitational forces for flight training (G-forces).
Used clinically for blood testing.
Page 7: Centripetal Force
Centripetal Force (Fc):
The net force causing uniform circular motion, directed towards the center.
Relation: Fc = m * ac = mv² / r
Newton's Second Law:
Fnet = ma applies to uniform circular motion: Fnet = mac.
Page 8: Example Problem 6.4
Knowns:
m = 900 kg, r = 500 m, v = 25.0 m/s
Unknowns:
a) Centripedal force, b) Minimum static coefficient of friction (ms)
Next Steps:
Draw forces.
Create a free-body diagram.
Use Newton’s Second Law to calculate.
Page 9: Solution to Example 6.4
Centripedal Force (Fc) Calculation:
Fc = mv² / r = (900 kg)(25.0 m/s)² / 500 m = 1125 N
Page 10: Friction Calculation
Static Friction:
Fnet in x-direction: Fc =
Fnet in y-direction: FN - mg = 0 (normal force balances weight).
Results lead to:
µs = Fc / (mg) = (Fc) / ((m)(g)).
Page 11-14: Example Problem 6.5
Banked Curve Calculation:
Knowns:
r = 100 m, θ = 65.0°
Objective:
Find speed (v) for a frictionless curve.
Steps include drawing free-body diagram and applying Newton’s Second Law.
Page 15: Lift Force on Jet
HW Problem #19:
Calculating lift force on a jet airplane in circular flight.
Given: m = 8910 kg, radius = 8.67 mi, period = 0.115 h.
Page 16-17: Fictitious Forces
Fictitious Forces:
Forces without physical origin in non-inertial frames.
Example: Centrifugal force during rotation.
Coriolis Force:
Causes moving objects to deflect observed from a rotating reference frame.
Page 18: Coriolis Effect
Coriolis Force in Nature:
Causes storm spirals:
Northern Hemisphere: counterclockwise.
Southern Hemisphere: clockwise.
Page 19: Newton’s Law of Gravitation
Universal Gravitation:
Every particle attracts every other particle: F = G(Mm / r²).
G = gravitational constant (6.674 x 10^-11 Nm²/kg²).
M = mass of one object, m = mass of the other, r = distance between them.
Page 20-22: Kepler’s Laws of Planetary Motion
First Law:
Orbits are ellipses with the Sun at one focus.
Second Law:
Imaginary line from Sun to planet sweeps equal areas in equal times.
Third Law:
The ratio of the squares of periods of two planets is equal to the ratio of the cubes of their average distances from the Sun.
Page 23: Extra Problems
Problems and additional exercises for further practice.
Page 24-28: Airplane and Ride Problem
Example of airplane radius, speed, and tension calculations.
Ride Problem:
Centripetal acceleration and normal force calculations for a Viking ship ride in an amusement park.
Includes problem-solving steps, free-body diagrams, and comparisons of forces experienced by a rider.