Circular Motion

Period and Frequency

• A CD rotates in a player at a constant speed of 240 rpm.– How frequently does it complete a revolution:

• In minutes?

• In seconds?

– How long does it take to complete one revolution?

Period and Frequency

• Period (T): The time it takes for an object in circular motion to make one complete revolution (or cycle)– Unit: s/rev– s/cycle– seconds

• Frequency (f): The number of revolutions, or cycles, in a given period of time– Equation:– Unit: 1/s – s-1

– Hertz (Hz)

Speed of an Object traveling in a Circle

• A car traveling in a circle with a radius of 40 m. – How far does it travel per cycle?

•

• The car completes 5 revolutions per minute.– What is its period? (How long does it take to complete a rev?)

•

• What is the car’s speed?

–

5 m

10 m

Each Revolution takes 2 seconds

What was the distance traveled by box 1? 2

1

What was the distance traveled by box 2?

What was the speed of box 1?

What was the speed of box 2?

Centripetal Acceleration

• Velocity is a Vector– Magnitude (i.e. size)– Direction

– Speed is constant, direction changing with time Velocity is changing with time, thus accelerating

v

vAfter 2 Secv

After 3 Sec

vAfter 6 Sec

After 5 Secv

After 4 Sec v

After 7 Sec

v

• Centripetal Acceleration: The acceleration of an object traveling in a circle at a constant speed– Speed is constant– Direction is changing– Acceleration vector points inward (center

seeking)

Centripetal Acceleration

v

ac

vf-vi

Centripetal Acceleration

• Why does the acceleration vector point inward?

vi

vf

vi-vi

vf

-vi

Centripetal Force

• ac, centripetal acceleration (m/s2)– “Center seeking”– Necessary to keep the object traveling in a circle

• Fc, centripetal force (N)

– Newton’s 2nd Law

• Newton’s 2nd Law for Circles

• NOTE: CENTRIFUGAL FORCE IS AN IMAGINARY FORCE TO EXPLAIN INERTIA! DOES NOT EXIST!

Centripetal Force

• The centripetal acceleration is caused by another force such as:– Friction– Tension– Normal Force– Gravity

Centripetal Force

• Steps to solve centripetal force problems:– Draw a free body diagram – Identify the force causing the centripetal force

• The force that is keeping the object going in a circle

– Set the causal force = centripetal force eq.– Solve for unknown

A 0.50 kg box is attached to string on a frictionless horizontal table. The box revolves in a circle of radius 2.8 m. If the box completes 1 revolution every 2.0 seconds, what is the tension in the string?

r

The centripetal force is caused by… the tension in the string

FN

Fg

FT

We know m & r, but need v

A 1200 kg car approaches a circular curve with a radius of 45.0 m. If the coefficient of friction between the tires and road is 1.20, what is the maximum speed at which the car can negotiate the curve?

r

FN

Fg

Ff

(Car is going into the page and turning right)

The centripetal force is caused by… friction