Rotational Motion and the Law of Gravity

7.1 Measuring Rotational Motion

  When something spins around a certain point its called –

circular motion

A circle of radius “r” has a circumference

2πr

angular measure of

360°

and has an 

Circular motion

is describes in terms of the angle object moves.

that the when the CD (compact disc) in Figure 1 rotates about its center—each point in the object follows a circular arc. Consider a line from the center of the CD to its edge. Each pit used to record sound along this line moves through the same angle in the same amount of time. The rotation angle is the amount of rotation and is analogous to linear distance. We define the rotation angle Δθ to be the ratio of the arc length to the radius of curvature:Δθ=Δsr.

  Angles in physics are measured in

radians

(rad) The angular displacement in radians (∆ϴ) is the ratio of the change in

arc length (s)

to the radius (r) of a circle

Formula:

∆ϴ =

∆s

r

Units cancel out and rad is used Conversions: 2 π radians = 360 ° = 1 rev And 1 rad = 180 0 / π = 57.3

0

Angular Speed describes the rate of rotation Formula:

ω = ∆ϴ / ∆ t

Units: rad / sec



Greek letter omega (ω) = angular speed

Average angular speed = angular displacement / time interval  When an object spins, we can describe how fast its going in terms of either: degrees/sec or revolutions/sec or

radians/sec Example Problem:

Earth makes one rotation (360 0 ) about its axis in one day (24 hours) If 360 0 = 2 π rad and 24 hours = 86400 sec And 2 x π = 6.28

Then, 2 π rad / 86400 sec = Earth’s angular speed = 7.27 x 10 -5 rad/sec

In a certain game show, contestants spin a wheel when it is their turn. One contestant gives the wheel an initial angular speed of 3.20 rad/s. It then rotates through one-and-one quarter revolutions and comes to rest on the BANKRUPT space. Through what angle has the wheel turned when its angular speed is 1.90 rad/s?

Angular acceleration occurs when angular speed changes Formula α = ∆ω ∆ t Greek letter α = alpha Average angular acceleration = change in angular speed / time interval

Comparing angular and linear quantities Linear Angular

x v a ϴ ω α

7.2 Tangential and Centripetal Acceleration

   Objects in circular motion have a tangential speed Any two objects have the same

angular speed and angular acceleration regardless of distance from center but

….

If the two objects are different distances from the axis of rotation, they have different

tangential speeds ( instantaneous linear speed at that point )

Tangential Speed

Formula: distance from the axis (

r

) x instantaneous angular speed (

ω)

V

tan

=

r

x

ω

tangential speed and angular speed

Tangential Acceleration

• instantaneous linear acceleration is tangent to the circular path

a

tan

= r x α

tangential acceleration = distance from the axis x angular acceleration

• •

Centripetal Acceleration

Centripetal “towards the center” Since Velocity is a vector there are 2 ways an acceleration • • • can be produced: change in

magnitude

and/or change in

direction

For a car moving in a circular path with constant speed the • object is accelerating due to a

change in direction

.

Experienced by any object that travels in a curved path

a

c

= v

tan 2

r

centripetal acceleration

=

(tangential speed) 2 distance from the axis

a c = r ● ω 2

Since V

tan

= r ω

avg

And a

c

= V

tan 2

r Sooo, a

c

= (r

2

● ω

2

) r Or

a

c

= r ● ω

2

7-3 Causes of Circular Motion

 Force that maintains circular motion

Centripetal Force

- any force towards center

Examples:

Earth’s gravitational pull on moon or the electric force that pulls electrons around atomic nuclei According to Newton’s 2 nd Law:

F

c

= m x a

c or

F

c

= m ● v

tan 2

r or

Centripetal force

F

c

= m x r x ω

2

Units are in Newtons (N)

3 Types of Acceleration

1)Linear (tangential) to change in speed

units: rad/s 2

– rate of change of speed… due to change in speed

units: m/s 2

2) Angular – rate of change of angular speed – due 3) Centripetal – a center-seeking acceleration – due to change in direction

units: m/s 2

http://www.pbs.org/opb/circus/cla ssroom/circus physics/centripetal-acceleration/

Newton’s Law of Universal Gravitation

Why do our planets stay in the sun’s orbit?

Why does the moon stay in orbit around the Earth?

 Mutual force of attraction between 2 objects  According to Newton’s 2nd Law of Motion

F g = G m 1 m 2 r 2 G = constant of universal gravitation

=

6.673 x 10 -11 N m 2 kg 2

Launch Speed less than 8000 m/s Projectile falls to Earth Launch Speed less than 8000 m/s Projectile falls to Earth Launch Speed equal to 8000 m/s Projectile orbits Earth - Circular Path Launch Speed greater than 8000 m/s Projectile orbits Earth - Elliptical Path