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:

  1. Draw forces.

  2. Create a free-body diagram.

  3. 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.