CIRCULAR MOTION

CIRCULAR MOTION

• Rotation along a circle, circular path, or circular orbit

• Can be uniform or non-uniform

UNIFORM CIRCULAR MOTION

• When an object is moving in a circle and its speed is constant

• Recall that speed is the magnitude of velocity

VELOCITY and ACCELERATION

• The direction of the object’s velocity is always tangent to the circle

• The direction of the motion is always changing

• The object is always ACCELERATING

VELOCITY and ACCELERATION

• As an object moves from point A to point B, its velocity changes from v1 to v2

• The direction of the acceleration is the same as the direction of the change in velocity (∆v=v2- v1) ∆v

v2

-v1

v1

AB

v2

CENTRIPETAL ACCELERATION

• The acceleration vector points directly towards the centre of the circle

• “Centre-seeking” or CENTRIPETAL ACCELERATION (ac)

• ac = v2/r

DERIVATION

r1=r2 because they are radii of the same circlev1=v2 because speed is constantr1 is perpendicular to v1 because the velocities are

tangent to the circleThe angle between corresponding members of sets of

perpendicular lines are equalSince the angles between the equal sides of 2 isosceles

triangles are equal, the triangles are similarThe two triangles can be used to find the magnitude of

acceleration.

DERIVATION

1. ∆r = ∆v r v2. The object travels from point A to point B in time

interval ∆t. ∆d= v∆t3. As the angle between A and B becomes very

small the length of ∆r becomes more nearly identical to the arc from A to B

∆r = ∆d

DERIVATION

4. Substitute ∆r for ∆d∆r = v∆t5. Substitute #4 into #1v∆t = ∆v r v6. Divide both sides by ∆tv= ∆vr v∆t

DERIVATION

7. Recall the definition for accelerationa = ∆v

∆t8. Substitute #7 into #6v = ar v9. Rearrange to solve for aa = v2

r

CENTRIPETAL FORCE

• The force causing centripetal acceleration always points towards the centre of the circular path

• It is not a “type of force” like friction or gravity

• It is a force that is required for an object to travel in a circular path

CENTRIPETAL FORCE• Centripetal force can be

supplied by any type of force.– Gravity provides the

centripetal force that keeps the moon in roughly a circular orbit

– Friction provides the centripetal force that causes a car to move in a circular path on a flat road

– Tension in a string tied to a ball will cause the ball to move in a circular path when you twirl it.

• Fc= mv2

r

DERIVATION

1. Recall Newton’s Second LawF = ma2. Recall the equation describing centripetal accelerationac = v2

r3. Substitute #2 into #1, omit the vector notation

because the Force and Acceleration always point towards the centre of the circular path.

Fc = mv2

r

Practice Problems

1. A car with a mass of 2135kg is rounding a curve on a level road. If the radius of the curvature is 52m and the coefficient of friction between the tires and the road is 0.70, what is the maximum speed at which the car can make the curve without skidding off the road?

Practice Problems

2. A yo-yo has a mass of 225g. The full length of the string is 1.2m. You decide to see how slowly you can swing it in a vertical circle while keeping the string fully extended, even when the yo-yo is at the top.

a)Calculate the minimum speedb)Find the tension on the string at the side and

bottom of the swing.

References

• http://upload.wikimedia.org/wikipedia/en/thumb/f/f1/Circular_motion_velocity_and_acceleration2.svg/512px-Circular_motion_velocity_and_acceleration2.svg.png

• http://www.dkimages.com/discover/previews/831/20114114.JPG

• http://en.wikipedia.org/wiki/Circular_motion