Download UNIFORM CIRCULAR MOTION Rotational Motion

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
nd
2
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
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