CIRCULAR MOTION

ROTATIONAL MOTION

• Objects that spin undergo rotational motion.

• Any point on the object has circular motion around the axis.

• The direction of motion is constantly changing.

ROTATIONAL QUANTITIES

- angular displacement –(degrees, radians, or revolutions)

= s/r - angular speed (rads/sec)

= /t - angular acceleration (rads/sec2)

= /t

1 radian = 57.3o 2 rads = 1 rev = 360o

Practice Problem

• While riding on a carousel that is rotating clockwise, a child travels through an arc length of 11.5 m. If the child’s angular displacement is 165o, what is the radius of the carousel?

Practice Problem

• A child at an ice cream parlor spins on a stool. The child turns counterclockwise with an average angular speed of 4.0 rads/sec. In what time interval will the child’s feet have an angular displacement of 8.0 rad?

Practice Problem

• The wheel on an upside down bicycle moves through 11.0 rad in 2.0 s. What is the wheel’s angular acceleration of its initial angular speed is 2.0 rad/s?

Tangential Speed

• Instantaneous linear speed• Varies with position from axis of

rotation• Speed along a line drawn tangent to

the circular pathvt = r

vt = 2r/TT is period (time/# revolutions)

Tangential Acceleration

• Tangent to circular path• Occurs when rotating objects change

speed • Example: A carousel speeds up

at = r

Practice Problem

• What is the tangential speed of a child seated 1.2 m from the center of a rotating merry go round that makes one complete revolution in 4.0 s?

• It takes 2.5 s for the merry go round to slow to a speed of .75 m/s. What is the tangential acceleration?

Centripetal Acceleration

• Occurs as object moves in a circular path because it changes direction

• Is constant or uniform• Directed toward the center

ac = vt2 ac = r2

r

Total Acceleration

• When both centripetal and tangential acceleration exist, at is tangent to circular path

• ac is toward the center

• Components are perpendicular

• atotal = square root of ac

2 + at2

• Direction = tan = ac/at

atotal

ac

at

Practice Problem

• The Polar Express has an angular acceleration of .50 rad/s/s. A rider sits 6.6 m from the center and makes 10 revolutions in 13 seconds. Find the tangential, centripetal, and total accelerations.

Circular Motion• object moves in a circular path • Continuous uniform acceleration• Ex: ball on the end of a string, Moon moving

about the Earth (almost circular)• Can be vertical or horizontal

animation by Behrooz Mostafavi

Vertical Circular motion

Fw

Fw

Ft

FtVmin occurs at the top

V = square root of rg

Ex: loop roller coaster

Practice Problem

• A ball of mass .45 kg is swung in a vertical circle. If the centripetal force on the ball is 12.5 N, what is the tension in the string at the top and bottom of the circle?

How do you feel…

• when sitting on the outside of the Polar Express ride at the State Fair

• when you are in a car that turns sharply to the left.

What causes these feelings?

Centrifugal Force – not real

Centripetal Force – the real force

• Explain how a bucket of water can be whirled in a a vertical circle without the water spilling out, even at the top of the circle when the bucket is upside down.

Horizontal Circular motion

Tension force acts horizontal and is constant.

If weight is small enough it can be ignored.

Ex: polar express

Centrifugal Forcethe fake force

What will happen…

• when a car doesn’t have enough friction force to get around a curve?

• If centripetal force is inward why will water not fall our of a cup that is swung in a vertical path?

What is required to cause the ball to have a curved path?

Centripetal Force• Required to maintain

centripetal acceleration (Newton’s Laws)

• Directed toward the center

• Acts at right angles to motion

• Ex: gravity, friction, strings…

Fc = mac = mr2 = mvt2/r

Practice Problem

• What would be the centripetal force on a 1500 kg car rounding a 8.5 m curve at a speed of 10.0 m/s?

It is sometimes said

that water is removed

from clothes in the spin

cycle by centrifugal force

throwing the water outward.

Is this correct?

The smaller the velocity of the object, the less centripetal force you will have to apply.

The smaller the length of rope (radius), the more centripetal

force you will have to apply to the rope.

The smaller the mass, the smaller the centripetal force (shown by the red vector labeled as the force of tension in

the rope, FT) you will have to apply to the rope.

If you let go of the rope (or the rope breaks) the object

will no longer be kept in that circular path and it will be free to

fly off on a tangent.

Newton’s Law of Gravitation

• Planets move in nearly circular orbits about the sun

• Gravity acts as the centripetal force.• Any two masses are attracted• Inverse square law

Fg = Gm1m2

r2

G = 6.67 x 10-11 Nm2/kg2

Practice Problem

• What is the force of attraction between you and the Earth?

Pictures and animations from

• http://regentsprep.org/Regents/physics/phys06/bcentrif/default.htm

• http://www.ap.smu.ca/demos/content/mechanics/waiters_tray/waiters_tray.html