Unit 3: Circular motion & Universal Gravitation
Overview
Circular motion describes the path an object takes around the center of motion. Objects have a centripetal acceleration towards the center of the circle, while the tangential speed of an object is perpendicular to the radius. Universal gravitation relates the force of gravity of a planet with relation to it's mass, an object's distance from the planet, and the radius of the planet.
Circular motion describes the path an object takes around the center of motion. Objects have a centripetal acceleration towards the center of the circle, while the tangential speed of an object is perpendicular to the radius. Universal gravitation relates the force of gravity of a planet with relation to it's mass, an object's distance from the planet, and the radius of the planet.
Circular Motion
Coordinate System
Cartesian Coordinate
- position is measured from the origin (x, y)
- horizontal x-axis with x-coordinate
- vertical y-axis with y-coordinate
Polar Coordinate
- position is measured with a radius and an angle (r, θ)
Polar to Cartesian Coordinates
- x = r cos θ
- y = r sin θ
Cartesian to Polar Coordinates
- r = square root (x^2 + y^2)
- θ = arctan (y/x)
Cartesian Coordinate
- position is measured from the origin (x, y)
- horizontal x-axis with x-coordinate
- vertical y-axis with y-coordinate
Polar Coordinate
- position is measured with a radius and an angle (r, θ)
Polar to Cartesian Coordinates
- x = r cos θ
- y = r sin θ
Cartesian to Polar Coordinates
- r = square root (x^2 + y^2)
- θ = arctan (y/x)
Angular Displacement & Velocity
We can determine the formula of an arclength using the angle and the radius. And the circumference of a circle is the same thing as the arclength of 360 degrees/one rotation. We can obtain the angular velocity by applying unit conversions onto Δθ over Δt. Eventually we will arrive at the units rotations per minute.
Tangential Speed
We can obtain tangential speed by multiplying the angular speed by the circumference ( w * 2πr)
Uniform Circular Motion
An object travels at a constant speed in a circular motion with a constant mass and radius.
We can determine the formula of an arclength using the angle and the radius. And the circumference of a circle is the same thing as the arclength of 360 degrees/one rotation. We can obtain the angular velocity by applying unit conversions onto Δθ over Δt. Eventually we will arrive at the units rotations per minute.
Tangential Speed
We can obtain tangential speed by multiplying the angular speed by the circumference ( w * 2πr)
Uniform Circular Motion
An object travels at a constant speed in a circular motion with a constant mass and radius.
Centripetal Acceleration
The centripetal acceleration is the acceleration towards the center of rotation. This can be calculated by using v^2/r. Centripetal acceleration can also be used to relate to ΣF and m. The formula a = ΣF / m allows us to determine the ΣF towards the center of the circle using the centripetal acceleration. ΣF = a * m = m * v^2 / r.
The centripetal acceleration is the acceleration towards the center of rotation. This can be calculated by using v^2/r. Centripetal acceleration can also be used to relate to ΣF and m. The formula a = ΣF / m allows us to determine the ΣF towards the center of the circle using the centripetal acceleration. ΣF = a * m = m * v^2 / r.
Universal Gravitation
Universal Gravitation exists between two objects with mass. The force of gravity depends on the mass of each object and the distance between the two objects.
The Law of Universal Gravitation
Gravity is proportional to the mass of each body and inversely proportional to the square distance between the two bodies.
Fg = G * (m1 * m2 / r^2)
G = Universal Gravitational Constant 6.67 * 10^-11 N * m^2 / kg^2
m1 = Usually Mp (mass of planet)
m2 = mass of object
r = radius of the planet
The Law of Universal Gravitation
Gravity is proportional to the mass of each body and inversely proportional to the square distance between the two bodies.
Fg = G * (m1 * m2 / r^2)
G = Universal Gravitational Constant 6.67 * 10^-11 N * m^2 / kg^2
m1 = Usually Mp (mass of planet)
m2 = mass of object
r = radius of the planet
Universal Gravitation Field
Gravitational field = Fg / m = G * Mp / (Rp)^2
G = Universal Gravitational Constant
Mp = mass of planet
Rp = radius of planet
Ex. Earth's Gravitational Field = G * Me / (Re)^2
Gravitational field = Fg / m = G * Mp / (Rp)^2
G = Universal Gravitational Constant
Mp = mass of planet
Rp = radius of planet
Ex. Earth's Gravitational Field = G * Me / (Re)^2