Circular Motion and Gravitation

13.1 Why Circular Motion Is Different from Linear Motion

In linear motion, acceleration changes the magnitude of velocity. In circular motion at constant speed, acceleration changes the direction of velocity while the magnitude stays constant. Both are accelerations — both require a net force by Newton’s second law — but they feel very different and require different analysis.

A key insight: an object moving in a circle at constant speed is always accelerating, even though its speedometer would read constant. The velocity vector is always tangent to the circle, and since the direction of a tangent changes continuously as the object moves around, the velocity is constantly changing. Any change in velocity is acceleration.

The acceleration required to maintain circular motion points toward the center of the circle at every instant. This is called centripetal acceleration. The corresponding force is the centripetal force.

13.2 Velocity in Circular Motion

For an object moving in a circle of radius r with constant speed v, the velocity vector is always tangent to the circle — perpendicular to the radius at every point. The speed v is the magnitude of this velocity vector and remains constant for uniform circular motion.

The period T is the time to complete one full circle. The speed is the circumference divided by the period:

v = 2πr / T

The frequency f is the number of complete circles per second, measured in hertz. Frequency and period are reciprocals:

f = 1 / T

13.3 Centripetal Acceleration

Even at constant speed, an object in circular motion has acceleration because its velocity direction changes. This centripetal acceleration always points toward the center of the circle.

a_c = v² / r

Equivalent expressions using period T or frequency f are:

a_c = 4π²r / T² = 4π²rf²

Centripetal acceleration increases with speed squared and decreases with increasing radius. Double the speed, and the centripetal acceleration becomes four times greater. Double the radius, and the centripetal acceleration becomes half as large.

13.4 Centripetal Force

By Newton’s second law, the centripetal acceleration requires a net force directed toward the center of the circle. This is the centripetal force:

F_c = mv² / r

This equation is not a new force law; it is Newton’s second law applied to circular motion. Whatever real force provides the centripetal acceleration — gravity, tension, friction, normal force, or any combination — its inward component must equal mv²/r for the object to maintain circular motion.

13.5 Sources of Centripetal Force

In different physical situations, different forces supply the centripetal acceleration. Recognizing the source of centripetal force is the first step in any circular motion problem.

13.6 Banking and Circular Motion Applications

A banked curve is tilted at an angle θ inward so that the horizontal component of the normal force provides centripetal acceleration, reducing or eliminating the need for friction. At the ideal banking angle for a given speed, no friction is needed at all.

tanθ = v² / (rg)

Vertical circular motion introduces an additional complication: the gravitational force changes its contribution to centripetal force depending on position in the loop. At the bottom of a loop, the normal force must exceed weight to provide centripetal acceleration upward. At the top of a loop, weight and normal force both point toward the center; the minimum speed occurs when N = 0.

13.7 Universal Gravitation

Gravity is universal: every object in the universe with mass attracts every other object with mass. This was Newton’s great insight, published in the Principia in 1687: the same force that causes an apple to fall causes the Moon to orbit Earth, and the same force governs planets orbiting the Sun.

Before Newton, gravity and the celestial motions of planets were considered entirely separate phenomena. Newton unified them with a single mathematical law, showed that the inverse-square relationship produced Kepler’s elliptical orbits, and demonstrated that the force causing free fall on Earth was exactly sufficient to keep the Moon in its orbit. This unification was one of the greatest intellectual achievements in scientific history.

13.8 Newton’s Law of Gravitation

The gravitational force between two masses is attractive, acts along the line joining them, and has magnitude:

F_g = Gm₁m₂ / r²

Key features of the law: the force is always attractive; it is proportional to each mass separately; it is inversely proportional to the square of the separation.

13.9 Gravitational Fields

Rather than thinking of gravity as an action-at-a-distance force, physicists often describe it using the concept of a gravitational field. The gravitational field g at any point in space is defined as the gravitational force per unit mass that would act on a small test mass placed there:

g = F_g / m = −GM / r² (pointing toward M)

Near Earth’s surface, the gravitational field is approximately uniform with magnitude g = 9.81 m/s², directed downward. This is the g used throughout Chapters 6–12. As altitude increases, g decreases:

g(r) = GM_Earth / r²

13.10 Weight vs. Mass Revisited

We can now revisit the weight-mass distinction with greater precision. The weight of an object is the gravitational force on it: W = mg, where g is the local gravitational field strength. Mass m is constant; weight varies with location because g varies.

An astronaut on the ISS, at 400 km altitude, experiences g ≈ 8.69 m/s². Their weight is W = m × 8.69 N — not zero. The astronaut appears weightless because the station and the astronaut are in the same free-fall orbit. Both accelerate toward Earth at the same rate, so there is no contact force between them. The normal force is zero, giving the sensation of weightlessness.

13.11 Orbital Motion

An orbit is a continuous free fall around a body. The satellite falls toward Earth due to gravity, but moves so fast horizontally that Earth’s surface curves away at the same rate the satellite falls. The result is a circular or elliptical path that never intersects the surface.

For a circular orbit, the gravitational force provides exactly the centripetal force needed:

GMm / r² = mv² / r

Solving for the orbital speed:

v_orbit = √(GM / r)

All objects at the same orbital radius orbit at the same speed, regardless of mass.

13.12 Satellites and Planetary Motion

The orbital period T is related to orbital radius r by combining v = 2πr/T with v = √(GM/r):

T² = (4π² / GM) r³

This relationship, first discovered empirically by Johannes Kepler in 1619 from astronomical observations, was later derived theoretically by Newton from his gravitational law. It is one of the great confirmations that Newton’s mechanics and gravity are correct.

13.13 Escape Velocity and Orbital Concepts

Escape velocity is the minimum speed required for an object to escape a planet’s gravitational pull entirely — to reach infinite distance without further propulsion. It is found by setting the object’s kinetic energy equal to the work needed to escape the gravitational well:

v_escape = √(2GM / r)

Note that v_escape = √2 × v_orbit: escape velocity is about 1.41 times the circular orbital speed at the same radius. For Earth’s surface, v_escape ≈ 11.2 km/s.

Geosynchronous orbit is the radius at which the orbital period equals Earth’s rotation period, 24 hours. A satellite in geosynchronous orbit above the equator appears stationary from the ground — this is the basis for communication and weather satellites. Geosynchronous radius is about 42,200 km from Earth’s center.

13.14 Common Mistakes in Circular Motion Analysis