Understanding Central Forces and Gravity
Central Forces and Angular Momentum
A central force acts along the position vector of the point where the force is applied. Gravity is a central force. Consider a particle of mass M (reference source) and a particle of mass m moving relative to M under the influence of gravity.
Angular Momentum
Angular momentum is the vector product of the position vector and linear momentum (mass times velocity).
Conservation of Angular Momentum
If the net torque is zero, angular momentum remains constant. Thus, the trajectories of planets lie in a fixed plane. When a particle moves around a point under the influence of a central force, its angular momentum is conserved.
Superposition Principle
To calculate the force on a mass due to a distribution of other masses, we sum the individual forces from each mass in the distribution.
Gravitational Field
From a gravitational perspective, the instantaneous appearance of forces between masses is difficult to reconcile with the finite speed of light. To address this, we introduce the concept of a gravitational field. A mass M creates a gravitational field, and any other mass m placed in this field experiences a force.
Gravitational Field Strength
Gravitational field strength is defined as the force per unit mass (Newtons/kg or m/s²). Near Earth’s surface, the field strength is approximately 9.8 m/s², which is the acceleration due to gravity. The gravitational field is a central field, as its direction always points towards the mass creating the field. If a mass “m” is placed in a region with multiple masses, it experiences a net force equal to the vector sum of the gravitational forces from each mass.
Kepler’s Laws
Kepler’s laws describe planetary motion:
- First Law: Planets move in elliptical orbits with the Sun at one focus.
- Second Law: The line joining a planet and the Sun sweeps out equal areas during equal intervals of time. This means a planet’s speed varies, being faster when closer to the Sun.
- Third Law: The cube of the semi-major axis of a planet’s orbit is proportional to the square of its orbital period.
Universal Gravitation
Galileo’s experiments showed that the rate of fall is proportional to time, the distance traveled is proportional to the square of time, and the rate of fall is independent of weight. Newton realized that the force causing objects to fall and the force keeping the Moon in orbit were the same. His Law of Universal Gravitation states: “The force between two masses is proportional to the product of the masses and inversely proportional to the square of the distance between them.” Eleven years later, Newton mathematically proved that a homogeneous sphere exerts the same gravitational force as if all its mass were concentrated at its center. Cavendish’s experiment determined the gravitational constant, allowing for the calculation of Earth’s mass.
Central Forces
A central force acts parallel to the position vector of the point where the force is applied. Gravity is a central force.
Superposition Principle
The net force on a mass due to a distribution of other masses is the vector sum of the individual forces.
Visualizing Gravitational Fields
Conditions for drawing gravitational field lines:
- Lines originate at infinity and terminate on masses.
- The direction of the field at any point is tangent to the field line at that point.
- Line density is proportional to field strength.
- Field lines cannot intersect; each point in the field has a unique direction.