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

Uniform Circular Motion

• 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

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

Centripetal Acceleration

• 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

• 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)

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

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 = vt

2 /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

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