Circular Motion and Gravitation

Chapter 5: Circular Motion; Gravitation

• This chapter covers kinematics and dynamics of uniform circular motion.
• It also explains highway curve design, non-uniform circular motion, Newton's Law of Universal Gravitation, gravity near Earth's surface, satellites, weightlessness, Kepler's Laws, and types of forces in nature.

5-1 Kinematics of Uniform Circular Motion

• Uniform circular motion is moving in a circle of constant radius at constant speed.
• Instantaneous velocity in circular motion is always tangent to the circle.
• The centripetal (radial) acceleration = v²/r, where v = velocity and r = radius.
• This acceleration is directed towards the center of the circle.

Period and Frequency

• Circular motion can also be described by frequency (f).
• Period (T) and frequency are related by T = 1/f.
• Speed v is distance/time = 2πr/T.

Example

• A 150-g ball at the end of a string revolves in a horizontal circle of radius 0.600 m at 2.00 revolutions per second.
• The speed is calculated as v = 2πr/T = 2π(0.600 m) / (0.500 s) = 7.54 m/s
• Centripetal acceleration is aR = v²/r = (7.54 m/s)² / (0.600 m) = 94.7 m/s².
• If the radius doubles to 1.20 m, but the period remains the same, the centripetal acceleration will change by a factor of 2.

Moon's Orbit

• The Moon's orbit has a radius of 384,000 km and a period of 27.3 days.
• The Moon's acceleration toward Earth is calculated by:
  • T = (27.3 d) (24.0 h/d)(3600 s/h) = 2.36 × 10⁶ s.
  • aR = v²/r = (2πr)² / T²r = 4π²r / T² = 4π²(3.84 × 10⁸ m) / (2.36 × 10⁶ s)² = 0.00272 m/s² = 2.72 × 10⁻³ m/s².

5-2 Dynamics of Uniform Circular Motion

• Maintaining uniform circular motion requires a net force.
• The net force is ΣFR = maR = mv²/r.
• The force is always inward, demonstrated by a ball on a string.
• Centrifugal force does not exist; the natural tendency of an object is to move in a straight line, but this is overcome in circular motion.
• If centripetal force disappears, the object moves off tangent to the circle.

Highway Curves

• Banking is the process of raising the outer edge of a curved surface to increase centripetal force for vehicles rounding curves.
• Banking helps reduce skidding.
• When tires slip, static friction becomes kinetic friction, which is bad.
• Kinetic friction has a smaller force than static friction.
• Kinetic friction opposes motion, making regaining control difficult.
• Banking curves can help keep cars from skidding.
• Every banked curve has a speed where the horizontal component of the normal force supplies the centripetal force with no friction needed.

Highway Curve Example

• A 1000-kg car rounds a flat curve of radius 50 m at 15 m/s.
• Determine if the car will skid in dry and icy conditions
• The net horizontal force needed to keep the car moving in a circle is (ΣF)R = maR = m v²/r = (1000 kg)(15 m/s)² / (50 m) = 4500 N.
• In dry conditions (µs = 0.60), max static friction force (Ffr)max = µsFN = (0.60)(9800 N) = 5880 N. Because 5880 N > 4500 N, the car follows the curve.
• In icy conditions (µs = 0.25), (Ffr)max = µsFN = (0.25)(9800 N) = 2450 N. Because 2450 N < 4500 N, the car will skid.

5-4 Non-uniform Circular Motion

• An object moving in a circular path but at varying speeds.
• Tangential acceleration is equal to the rate of change of the magnitude of the object's velocity.
  • atan = Δv / Δt.
• Radial (centripetal) acceleration changes the direction of the velocity.
  • aₐᵣₐ = v²/r.
• The total vector acceleration is the sum of atan + arad
• Magnitude of total acceleration is √(aₐ² + aᵣ²).

Non-uniform Example

• A race car starts from rest and reaches 35 m/s in 11s on a 500 m radius track.
  • The tangential acceleration is atan = Δv / Δt = (35 - 0) / 11 = 3.2 m/s².
  • At 15 m/s, radial acceleration is arad = v²/r = 15² / 500 = 0.45 m/s².
  • Total acceleration magnitude is a = √(3.2² + 0.45²) = 3.23 m/s².

5-5 Newton's Law of Universal Gravitation

• Gravity is the same force that keeps the Moon orbiting the Earth.
• Earth exerts a force on you while you exert one on the Earth.
• The gravitational force is proportional to both masses.
• Newton concluded that gravitational force decreases as the inverse square of the distance between masses when observing planetary orbits.
• Law of Universal Gravitation:
  • FG = G (m1m2 / r²), where G = 6.67 × 10⁻¹¹ N·m²/kg².
• The magnitude of the gravitational constant G can be measured.
• This can be shown with the Cavendish experiment.

Example

• A 50-kg person is sitting on a bench 50 cm apart from a 70-kg person.
• Gravitational force calculated as FG = G(m1m2 / r²) = 6.67 × 10⁻¹¹ × 50 × 70 / 0.5² = 1 × 10⁻⁶ N

Orbits

• A 2000-kg spacecraft orbiting at two Earth radii means that:
  • r = 2rE = 2 × 6.4 × 10⁶ = 1.28 × 10⁷ m.
  • m₁ = 2000 kg and m₂ = 6.0 × 10²⁴ kg.
  • Force of gravity FG = (6.67 × 10⁻¹¹ × 2000 × 6.0 × 10²⁴) / (1.28 × 10⁷)² = 4885 N.

5-6 Gravity Near the Earth's Surface

• mg = G (mM / rE²).
• The local gravitaional acceleration constant:
  • g = G (mM / rE²).
• Mass of the Earth is calculated as:
  • mE = (grE²) / G = (9.80 m/s²)(6.38 × 10⁶ m)² / (6.67 × 10⁻¹¹ N·m²/kg²) = 5.98 × 10²⁴ kg.
• Acceleration due to gravity varies with altitude, local geology, and Earth's shape
• On Earth: -New York = 9.803 m/s² -San Francisco = 9.800 m/s² -Denver = 9.796 m/s² -Pikes Peak = 9.789 m/s² -Sydney, Australia = 9.798 m/s² -Equator = 9.780 m/s² -North Pole = 9.832 m/s²

5-7 Satellites and "Weightlessness"

• Tangential speed must be high enough to not return to Earth, but not escape its gravity.
• Satellites are kept in orbit. Satellite are continually falling but the Earth curves from underneath it.
• Gravitational force is balanced by the centripetal force.
• v = √(GM / r) and r = rE + h (height of satellite)
• Satellites with different masses orbiting Earth at the same distance have the same speeds and the same period.

Weightless in Orbit?

• In orbit their is a feeling described as weightlessness.
• This feeling is created as the satellite and its contents are in free fall, so there is no normal force.
• It is more accurately called apparent weightlessness, as gravitational force remains.
• Feeling of weightlessness can also be briefly felt on Earth in free fall.

Gravitational Potential Energy

• PE is the negative work done by gravitational force during displacement.
• Formula: ΔU = Uf - Ui = -∫F(r)dr from ri to rf. Can simplify to to Uf-U₁=-GmEm (1/rf - 1/ri)
• When ri = ∞, then U = 0. Making the formula U = -G Mm / r or U = -G Mm / rE+h
• Total: Utotal = -G (m1m2/r12 + m1m3/r13 + m2m3/r23)

Escape Velocity

• The minimum speed needed at a planet's - Earth's - surface to escape its gravitational influence
• Vesc =√(2GM / rE) or √(2 g rE)

Kepler's Laws

• Kepler's laws describe planetary motion.
  • (1) Planet orbits are ellipses with the Sun at one focus with constant aera.
  • (2) An imaginary line from each planet to the Sun sweeps equal areas in equal times.
  • (3) The ratio of the square of a planet's orbital period is proportional to the cube of its mean distance from the Sun.
    • Formula: (T1/T2)^2 = (r1/r2)^3 and T^2=kr^3
    • k = 4π²/GM☉