ROTATIONAL MOTION Uniform Circular Motion

Uniform Circular Motion Riding on a Ferris wheel or carousel  Once a constant rate of rotation is reached (meaning the rider moves in a circle at a constant speed)  UNIFORM CIRCULAR MOTION Recall Distinction: Speed – Magnitude or how fast an object moves Velocity – Includes both magnitude AND direction Acceleration – Change in velocity Preview Kinetic Books- 9.1 Other exs- Satellite in circular orbit around the Earth; Toy train on a circular track 9.1 Even as it moves around the curve at a constant speed, its velocity constantly changes as its direction changes. A change in velocity is called acceleration, and the acceleration of a car due to its change in direction as it moves around a curve is called centripetal acceleration. Although the car moves at a constant speed as it moves around the curve, it is accelerating. This is a case where the everyday use of a word − acceleration − and its use in physics differ. A non-physicist would likely say: If a car moves around a curve at a constant speed, it is not accelerating. But a physicist would say: It most certainly is accelerating because its direction is changing. She could even point out, as we will discuss later, that a net external force is being applied on the car, so the car must be accelerating.

Uniform Circular Motion Motion in a circle with constant speed “Uniform” refers to a constant speed Velocity is changing though! Length of the velocity vector does not change (speed stays constant), but the vector’s direction constantly changes Since acceleration = Change in velocity, the object accelerates as it moves around the track Instantaneous velocity is always tangent to the circle of motion

Uniform Circular Motion Period Amount of time to complete one revolution Period for uniform circular motion T = 2πr/v (2πr Distance around circle = circumference) T = period (s) r = radius (m) v = speed (m/s) π = 3.14 Complete Kinetic Books 9.2- Ex 1

Uniform Circular Motion Tangential speed (vt) An object’s speed along an imaginary line drawn tangent to the object’s circular path Depends on the distance from the object to the center of the circular path Consider a pair of horses side-by-side on a carousel Each completes one full circle in the same time period but the outside horse covers more distance and therefore has a greater tangential speed Do kinetic books interactive checkpoint 9.3 r = 0.06 m; T = 60s/1 min x 1 min/205 rev = 0.29s  v = 2(3.14)(0.06)/0.29 = 1.3 m/s

Centripetal Acceleration Acceleration due to change in direction in circular motion In uniform circular motion, acceleration = CONSTANT Points toward the center of the circle  perpendicular to the velocity vector Train goes around a track at a constant speed Train’s velocity is changing because it is changing direction Change in velocity = Acceleration

Centripetal Acceleration Points toward the center of the circle ac = vt2 /r ac = Centripetal acceleration (m/s2) vt = Tangential speed (m/s) r = radius of circular path (m) Complete Kinetic books 9.4 Example 1

Problem A car moves at a constant speed around a circular track. If the car is 48.2 m from the track’s center and has a centripetal acceleration of 8.05 m/s2, what is the car’s tangential speed? ac = vt2 / r  vt = √acr  vt = √(8.05 m/s2)(48.2m) vt = 19.7 m/s

Centripetal Force Forces & Centripetal Acceleration Yo-yo swings in a circle  it accelerates, because its velocity is constantly changing direction In order to have centripetal acceleration there must be a force present on the Yo-yo Force that causes centripetal acceleration points in the same direction as the centripetal acceleration  Toward the center of the circle FIRST!!! Show concept 1 in Kinetic Books 9.6- then display info on this slide

Centripetal Force Any force can be centripetal Yo-yo moves in a circle by the tension force in the string Gravitational force keeps satellites in circular orbits When forces act in this fashion, both tension and gravity  Centripetal forces

Newton’s 2nd Law Newton’s 2nd Law F = ma When objects move in a circle  Centripetal acceleration ac = vt2 /r …Now, plug this into F = ma CENTRIPETAL FORCE (Fc): Fc = m (vt2/r) Fc = Newton m = mass (kg) vt = tangential speed (m/s) r = radius of the circular path (m) Force points toward the center of the circle * Preview Kinetic Books 9.6  Equation 1 section & have students complete example 1

Problem A pilot is flying a small plane at 56.6 m/s in a circular path with a radius of 188.5 m. The centripetal force needed to maintain the plane’s circular motion is 1.89 x 104 N. What is the plane’s mass? Fc = mvt2 / r m = Fc r / vt2 = (1.89 x 104 N)(188.5 m)/(56.6 m/s)2 m = 1110 kg

Centripetal Force Centripetal Force Acts at right angles to an object’s circular motion Necessary for circular motion