Uniform Circular Motion (UCM) The object travels in a circular path with a constant speed. Its velocity is tangent to the circle and is changing due to.

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Uniform Circular Motion (UCM) The object travels in a circular path with a constant speed. Its velocity is tangent to the circle and is changing due to the changing direction. The unbalanced force is always directed toward the center of the circle. The acceleration is always directed toward the center of the circle.

v v v v v v v v F F F F F F F F Without a force toward the center the object will fly off on a line tangent to the circle.

Something you should already know: The Circumference (C) of a circle: By the way the area is given by: A

New Definition: T = Period = The time for one cycle. (measured in seconds) In the case of circular motion the period is the time for one cycle, which is the time for one complete revolution.

What are the units for an angle: θ 0º0º 270º 180º 90º θ = 60º Degrees Revolutions θ = 1/6 1/2 1/4 0 3/4 Radians 0 π/2 π 3π/2 2π2π θ = π/3 = 3.14/3 ≈ 1 rad s =1radian

Linear Speed vs. Angular Speed V = Linear Speed – The rate at which the object travels a distance. ω = Angular Speed – The rate at which the object travels through an angle. Units: m/s Units: deg/s, rad/s, Revolutions per minute (rpm)

Relationship between linear and angular quantities: s = arc length (m) r = radius (m) v = linear speed (m/s) ө = angle (radians) ώ = angular speed (radians/s)

Two objects travel in a circular path. Object A has a radius of 0.8 meters and object B has a radius of 2 meters. They both complete a revolution in 5 seconds. Compare their linear and angular speeds. A B.8m 2m Object A Object B

Let’s do some conversions: 80rpm = ______ m/s radius = 40cm ? º = _____ radians ? 120 deg 2.1

An object traveling in a circle with a constant speed is still accelerating due to its changing direction. This acceleration is called the centripetal acceleration and is given by the following equation: Units:

The steps to solving a problem: Draw a force diagram Apply Newton’s 2 nd Law Substitute the centripetal acceleration equation for the acceleration.

Example 1 A 300g ball is swung on a string in a horizontal circle. (Neglect gravity) The mass has an angular speed of 120rpm. Find the tension in the string. 3m m=300g FTFT

Example 2: A car traveling with a constant speed rounds a circular curve of radius 15 meters. The coefficient of friction between the road and the tires is 0.8. What is the maximum speed the car can have and still successfully negotiate the curve? 15m Top view v f

Example 3: A 500kg satellite is in a geosynchronous orbit. (It takes 24 hours to orbit the earth one time.) The satellite has a orbital radius of 42000km. What is the force of gravity on the satellite? r = 42000km FgFg