Circular Motion

Notes

(Ref p234-239 HRW)

Examples of circular motion

Uniform Circular Motion

Definition • Motion exhibited by a body following a circular path at

constant (or uniform) speed, with characteristics of …

Frequency (f) – “how often” • # of cycles, rotations or revolutions per unit time.

• SI units are cycles/sec or hertz (Hz)

Period (T) – “how long” • Time taken for 1 complete rotational cycle or

revolution.

• SI units are seconds.

Frequency-Speed relationship • f = 1/T

Example

Q: A merry-go-round takes 5 seconds to

make 1 complete revolution.

• What is the frequency (f) of rotation?

A: Given T, find f = 1/T

• T = 5 seconds

• f = 1/T = 1/5

• f = 0.2 rev/sec (or cycles/sec)

Example 2

Q: A CD rotates at 500 rpm near the

center.

• What is the period (T) of the rotation?

A: Given f, find T = 1/f

• f = 500 rpm

• T = 1/f = 1/500 = 0.002 minutes

• T = 0.12 sec (= 0.002 x 60)

• T = 120 milliseconds (ms)

Linear (tangential/orbital) velocity

v = distance traveled around a circle per

unit time (vav = dist/time)

• vav = circumference/period

• vav = 2(radius)/period

• vav = 2r/T

Direction of velocity vector is tangential

(perpendicular) to radius

Units are m/s

Example – tether ball

What is the orbital speed (v) of a tennis ball

that is being swung on the end of a 1.5 m

(r) string and takes 1.25 seconds (T) for a

complete revolution?

Given: r = 1.5m, T = 1.25 s

Solve v = 2(radius)/period

• v = 2 (1.5)/1.25

• v = 7.54 m/s

Centripetal Acceleration (ac)

ac = linear velocity2

radius of rotation

• ac = v2/r

Direction of acceleration

vector is inward along

radius (center seeking).

Units are m/s2

Acceleration = rate of change of velocity, so…

Example

What is the centripetal acceleration (ac) of the tennis ball in the previous example?

Solve ac = v2/r (v = 7.54 m/s, r=1.5) • ac = 7.542/1.5

• ac = 37.89 m/s2

Per Newton, if there is acceleration there must be a non-zero…

Centripetal force (Fc)

Fc = mass x ac (Newton’s 2nd Law)

Fc = mac = mv2/r

Direction of force vector is inward along

radius (center-seeking).

Units are Newtons (N)

Example – Merry Go Round

A merry-go-round has a diameter of 10m

and a period of 10 seconds. There are 2

circles of horses, at 3 m radius and at

the outer edge. If a rider has a mass of

50 kg, calculate:

• Speed of the horses at the edge and 3 m

• Centripetal acceleration of each horse

• Centripetal force on each horse

Solution – Merry Go Round

v(edge) = 2π r/T = 2π(5)/10 = 3.14 m/s

v(3 m) = 2π r/T = 2π(3)/10 = 1.88 m/s

ac (edge) = v2/r = 3.142/5 = 1.972 m/s2

ac (3m) = v^2/r = 1.882/3 = 1.178 m/s2

Fc(edge) = mac = 50(1.972) = 98.6N (inward)

Fc (3m) = mac = 50(1.178) = 58.9N (inward)

Circular Motion Summary

Torque ()

Turning force or “moment” of a force

Product of the perpendicular component of a force applied at a distance from the axis of rotation (pivot or fulcrum) of an object. • = Force┴ x lever arm

• = F┴ x L

Units are N-m.

Convention direction • Counterclockwise (ccw) = +ve

• Clockwise (cw) = -ve

pivot

Torque Examples

Torque – Change angle and lever

arm

Models

2 models for analysis

• Single pivot (ex. see-saw)

• Double pivot (ex. plank,

stretcher, bridge)

Single Pivot (Seesaw) Model

To achieve balance

• total ccw torque = total cw torque

• total means all objects contributing to the

respective torques.

• ccw = cw

W1 W2

d

1

d2

Center of Gravity (CoG)

The point on an object through which all

the mass is deemed to be acting.

• The object is assumed to be uniform in

structure

Important concept when considering

double pivot models

Pondering…What happens to the plane if the CoG moves backward?

Double Pivot (Bridge/Plank)

Model

For balance,

• the sum of the CW torques = the sum of the

CCW torques

CCW CW

FL

Mass of plank Mass of box

FR mass

Torque Practice

Overheads…

Practice – where should the girl

sit?

Moment of Inertia or Rotational

Inertia (I)

Defn: The tendency of a body rotating about a fixed axis to resist change in rotational motion. • Rotational equivalent of mass. • Dependent on the distribution of the mass of the

object (where is the mass located?)

Formula • I = “shape” constant x mass x linear dimension

(squared) • Where shape is a numeric multiplier dependent upon the

object shape

• I = (k)mr2

• see text (P103) for models

Units • kg-m2

Symbol

Models

Rotational Inertia Models (P103)

Note: cylinder = disk!

Example

What is the moment of inertia of of a disk-

shaped wheel that has a mass of 15 kg

and a diameter of 2.8 m?

Solve: Refer to formulas on p 103

• Model = cylinder > I = ½ mr2

• I = ½ (15)(2.8/2)2

• I = 14.7 kg.m2

Angular Momentum (L)

A measure of the how much a rotating body resists stopping. • Rotational equivalent of linear momentum

L = mass x linear velocity x radius of rotation • L = mvr

Units • kg.m/s.m > kg.m2/s

Symbol

Example of Angular Momentum

Jupiter orbits the sun with a speed of

2079 m/s and orbit of 71,398,000 m from

the sun. If Jupiter’s mass is 1.9 x 1027 kg,

what is its angular momentum (L)?

Solve: L = mvr

• L = 1.9 x 1027 x 2,079 x 71,398,000

• L = 2.82 x 1038 kg.m2/s

Conservation of Angular

Momentum

Linear momentum is conserved, so…

Angular momentum is conserved across an event

Visualize an ice skater doing pirouettes…

• Skater has arms out (larger radius↑) – smaller spinning velocity↓

• Skater pulls her arms in (smaller radius↓)– greater spinning velocity↑.

• L before = Lafter (Conservation of Ang Momentum)

• (mvr)before = (mvr)after

• v1r1 = v2r2 (mass (m) cancels)

Skater Example of Cons of

Angular Momentum

v * R V * r

v*r product does not change across an event

Before After

Example of Conservation of

Angular Momentum

A physics student is spinning around in a chair @ 1.5 m/s with his arms stretched out 0.6 m from the center of his body, holding in each hand a 2 kg mass. If he pulls in his arms to 0.2 m from the center of his body, how fast does he now spin?

Solve: Conservation of Angular momentum • Angular momentum (before) = angular momentum (after)

• v1 x r1 = v2 x r2

• 1.5 x 0.6 = v2 x 0.2

• v2 = 0.9/0.2

• v2 = 4.5 m/s

Linear vs Rotational Model

Parameter Linear Rotational

Inertia Mass Moment of Inertia

Forces Force Torque

Momentum Momentum Angular momentum