Arc Length Formula

1
Arc Length Formula

• Pre-Calculus
• Unit 4, Day 5

2
Arc Length and Central Angles
3
Example

• Find the measure of a rotation in radians when a
  point 2 m from the center of rotation travels 4 m.

4
Example

• Find the length of an arc of a circle of radius 5
  cm associated with an angle of ?/3 radians.

5
Linear Angular Velocity

• Things that turn have both a
  linear velocity and an angular velocity.

www.vcsc.k12.in.us/lincoln/math/CaseNotes/.../Line
arAngVel.ppt
6
Linear Velocity
Definition
Linear Velocity is distance/time
Distance
Linear Speed
Time
7
Angular Velocity
Definition
Angular Velocity is turn/time
Rotation in radians
Angular Speed (omega)
Time
8
Linear Angular Velocity
Recall Arc Length Formula?
Linear Velocity in terms of Angular Velocity
9
Let us take 2 pendulums hung on a slim rotating
rod for analysis.

• If the 2 pendulums (A and B) rotate one full
  cycle, the time taken by them is the same.They
  covered the same amount of angular distance (360
  degree) within the same amount of time.This
  showed that they have exactly the SAME angular
  speed.But is the Linear speed the same?

http//mathsisinteresting.blogspot.com/2008/08/lin
ear-angular-speed.html
10
Let us take 2 pendulums hung on a slim rotating
rod for analysis.

• The length of the 2 circumferences travelled by
  the individual pendulums are not the same.The
  linear length or distance is therefore NOT the
  same.Length 2 x (pi) x radius.They took the
  same time to complete one full cycle,
  though.The linear speed is thus DIFFERENT,
  having travelled different length for the same
  amount of time.

http//mathsisinteresting.blogspot.com/2008/08/lin
ear-angular-speed.html
11
Example

• A satellite traveling in a circular orbit
  approximately 1800 km. above the surface of Earth
  takes 2.5 hrs. to make an orbit. The radius of
  the earth is approximately 6400 km.
• a) Approximate the linear speed of the
  satellite in kilometers per hour.
• b) Approximate the distance the satellite
  travels in 3.5 hrs.

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Example

• r 6400 1800 8200
• t 2.5 hrs.
• a) Approximate the linear speed of the
  satellite in kilometers per hour.

13
Example

• r 6400 1800 8200
• t 2.5 hrs.
• b) Approximate the distance the satellite
  travels in 3.5 hrs.

14

• A small pulley 6 cm in diameter is connected by a
  belt to a larger pulley 15cm in diameter. The
  small pulley is turning at 120 rpm.
• Find the angular velocity of the small pulley in
  radians per second.
• Find the linear velocity of the rim of the small
  pulley.

15

• A small pulley 6 cm in diameter is connected by a
  belt to a larger pulley 15cm in diameter. The
  small pulley is turning at 120 rpm.
• Find the angular velocity of the small pulley in
  radians per second.

16

• A small pulley 6 cm in diameter is connected by a
  belt to a larger pulley 15cm in diameter. The
  small pulley is turning at 120 rpm.
• Find the linear velocity of the rim of the small
  pulley.

17
Homework

• Page 527
• 65-68, 71-79