U11: Uniform Circular Motion and Gravitation

When objects rotate about some axis, each point in the object follows a circular arc

● The rotation angle is the ratio of the arc length to the radius of curvature

○ 𝛥𝜃 = 𝛥𝑠/𝑟

■ The arc length is the distance traveled along a circular path

■ r is the radius of curvature of the circular path

● The result is the basis for defining the units to measure rotation angles, 𝛥𝜃 to be radians

○ 2𝜋radians = 1 revolution

Angular velocity is the rate of change of an angle

● 𝑤 = 𝛥𝜃/𝛥𝑡

○ The greater the rotation angle in a given amount of time, the greater the angular velocity

○ The unit for angular velocity is radians per second

Angular velocity is analogous to linear velocity

● 𝑣 = 𝛥𝑠/𝛥𝑡

● 𝑣 = (𝑟𝛥𝜃)/𝛥𝑡 = 𝑟𝑤

○ 𝑤 = 𝑣/𝑟

Centripetal acceleration is the acceleration of an object moving in uniform circular motion (resulting from a net external force)

● Centripetal means “toward the center” or “center seeking”

● 𝑎c = (𝑣2)/𝑟 = 𝑟𝑤2

● A centrifuge is a rotating device used to separate specimens of difference densities

○ High centripetal acceleration significantly decreases the time it takes for separation to occur

Centripetal force is any net force causing uniform circular motion

● The direction of a centripetal force is toward the center of curvature (the same as the direction of centripetal acceleration)

● It is always perpendicular to the path

● 𝐹𝑐 = 𝑚𝑎𝑐 = 𝑚(𝑣2/𝑟) = 𝑚𝑟𝑤2

○ For a given mass and velocity, a large centripetal force causes a small radius of curvature (a tight curve)

In banked curves, the slope of the road helps negotiate the curve

● The greater the angle, the faster you can take the curve

● For ideal banking, the net external force equals the horizontal centripetal force in the absence of friction

Newton’s universal law of gravitation states that every particle in the universe attracts every other particle with a force along a line joining them

● The force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them

● We assume that the body acts as if its entire mass is concentrated at one specific point called the center of mass

● 𝐹 = 𝐺(𝑚𝑀/𝑟2)

○ 𝑔 = 𝐺(𝑀/𝑟2)

■ G is a proportionality factor called the gravitational constant (it is given on your reference sheet)

These are Kepler’s three laws of planetary motion:

● The orbit of each planet about the Sun is an ellipse with the sun at one focus

● Each planet moves so that an imaginary line drawn from the sun to the planet sweeps out equal areas in equal times

● The ratio of the squares of the periods of any two planets about the Sun is equal to the ratio of the cubes of their average distances from the Sun