Circular motion

Angle in radian

arc length/ radius

Angle in radius (terms of pi)

2π r / r = 2π (whole circle)

angular velocity (a.k.a angular frequency)

ω = θ/ t (s= vt, v = θ)
(measured in rad s ^-1)

Angular velocity for 1 circle

ω = 2π/ t = 2πf

Linear speed

v = r ω

Centripetal

Towards the centre of the circle

radians to degrees

x 180/ π

degrees to radian

π/ 180 x angle in degrees

1 radian

57 degrees

Centripetal (Defining condition)

• if the resultant force acting is perpendicular to the velocity of the object

• the resultant force acting towards the centre is the centripetal force (the velocity acts tangentially)

Energy change

• centripetal force and velocity are perpendicular = no work done

• energy of the object remains constant

• velocity constantly changes

• object accelerates towards the centre of the circle

points about circular motion

• linear speed (v) is constant

• direction changes as object goes round circle

• velocity changes

• accelerates towards centre of circle

• force towards centre F is the resultant force already present

acceleration

a = v²/ r or a= w² r

1 radian (definition)

angle subtended at the centre of a circle by an arc in length to the adius

What is angular velocity?

The angle (in radians) turned through in one second

units = rad s ^-1

frequency f def

number of revolutions per second

units = hertz

v=

ω=

r =

v = linear speed

ω= angular velocity

r = radius

angle θ radians defined as ratio of arc length to radius:

θ = arc length/ r

Calculate the angular velocity of a Ferris wheel that rotates 3 times in 2 minutes.

step 1: formula

ω = θ/ t

step 2: calc. change in angle in radians:

θ = 3(rotations) x 2π = 6π radians

step 3: convert minutes to seconds: 120 seconds

ω = 6π / 120 = 0.157 rads ^-1

relating frequency, time and angular velocity

ω = 2πf

(f = 1/T)

ω = 2π/ T

Determine the angular velocity (ω) of a ceiling fan completing 120 revolutions per minute.

Step 1: Convert rpm to Frequency (f)

to convert from rpm to frequency, divide by 60

120 rpm = 2 rev s^-1

Step 2: Formula

ω = 2 π x f

ω = 2π x 2 = 4π rad s ^-1

2 equations for centripetal acceleration

a(c) = v² / r

a(c) = ω² x r

Increasing the radius leads to

…longer revolution times while maintaining the same centripetal force.

simple harmonic motion

The acceleration (a) of the object is proportional to the displacement (s) and acts in the opposite direction to the displacement.

gradient of acceleration and displacement graph:

negative gradient passing through both -x and -y

The acceleration in Simple harmonic motion is related to the displacement by the equation:

a = - ω² x

(x = displacement from midpoint)