Set 4 Set 4 Circles and NewtonCircles and Newton

February 3, 2006February 3, 2006

Where Are WeWhere Are We• Today

– Quick review of the examination– we finish one topic from the last chapter – circular motion

• We then move on to Newton’s Laws• New WebAssign on board on today’s lecture material

– Assignment – Read the circular motion stuff and begin reading Newton’s Laws of Motion

• Next week– Continue Newton– Quiz on Friday

• Remember our deal!

Remember from the past …Remember from the past …• Velocity is a vector with magnitude

and direction.• We can change the velocity in three

ways– increase the magnitude– change the direction– or both

• If any of the components of v change then there is an acceleration.

Changing VelocityChanging Velocity

v1

v2v2

va

Uniform Circular MotionUniform Circular Motion• Uniform circular motion occurs when an

object moves in a circular path with a constant speed

• An acceleration exists since the direction of the motion is changing – This change in velocity is related to an

acceleration

• The velocity vector is always tangent to the path of the object

Quick Review - RadiansQuick Review - Radians

s

Radians r

s

Changing Velocity in Changing Velocity in Uniform Circular MotionUniform Circular Motion

• The change in the velocity vector is due to the change in direction

• The vector diagram shows v = vf - vi

The accelerationThe acceleration

2

r

va

r

v

t

tvr

at

vt

v

vv

CentripetalAcceleration

Centripetal AccelerationCentripetal Acceleration• The acceleration is always

perpendicular to the path of the motion

• The acceleration always points toward the center of the circle of motion

• This acceleration is called the centripetal acceleration

Centripetal Acceleration, Centripetal Acceleration, contcont

• The magnitude of the centripetal acceleration vector was shown to be

• The direction of the centripetal acceleration vector is always changing, to stay directed toward the center of the circle of motion

2

C

va

r

PeriodPeriod• The period, T, is the time required

for one complete revolution• The speed of the particle would be

the circumference of the circle of motion divided by the period

• Therefore, the period is

2 rT

v

Tangential AccelerationTangential Acceleration• The magnitude of the velocity could

also be changing• In this case, there would be a

tangential acceleration

Total AccelerationTotal Acceleration• The tangential

acceleration causes the change in the speed of the particle

• The radial acceleration comes from a change in the direction of the velocity vector

Total Acceleration, Total Acceleration, equationsequations

• The tangential acceleration:

• The radial acceleration:

• The total acceleration:– Magnitude

t

da

dt

v

2

r C

va a

r

2 2r ta a a

Total Acceleration, In Terms Total Acceleration, In Terms of Unit Vectorsof Unit Vectors

• Define the following unit vectors

– r lies along the radius vector

is tangent to the circle

• The total acceleration is

ˆˆ andr

2ˆ ˆt r

d v

dt r

va a a r

A ball on the end of a string is whirled around in a horizontal circle of radius 0.300 m. The plane of the circle is 1.20 m above the ground. The string breaks and the ball lands 2.00 m (horizontally) away from the point on the ground directly beneath the ball's location when the string breaks. Find the radial acceleration of the ball during its circular motion.

12

2

rr

v

A pendulum with a cord of length r = 1.00 m swings in a vertical plane (Fig. P4.53). When the pendulum is in the two horizontal positions = 90.0° and = 270°, its speed is 5.00 m/s. (a) Find the magnitude of the radial acceleration and tangential