Lecture Notes: Area of Sector, Linear Speed, and Angular Speed

Area of a Sector of a Circle

• Formula: The area ( A ) of a sector of a circle with radius ( r ) and central angle ( \theta ) in radians is given by: [ A = \frac{1}{2} r^2 \theta ]
• Important Note: ( \theta ) must be in radians. Convert degrees to radians when necessary.
  • Conversion: Multiply degrees by ( \frac{\pi}{180} ).

Example Calculation

• Problem: Find the area of a sector with:
  • Radius: 5 feet
  • Central angle: 40 degrees
• Steps:
  1. Convert 40 degrees to radians:
     • ( \frac{40 \pi}{180} = \frac{2 \pi}{9} ) radians
  2. Use the formula:
     • ( A = \frac{1}{2} \times 5^2 \times \frac{2 \pi}{9} )
     • Rounded to two decimal places: 8.73 square feet

Linear Speed

• Concept: Relates to the distance around a circle (arc length).
• Formula: [ v = \frac{s}{t} ]
  • ( v ): Linear speed
  • ( s ): Arc length
  • ( t ): Time
• Relation to Distance Formula: ( s = v \cdot t )

Angular Speed

• Concept: Measured in radians per unit time.
• Formula: [ \omega = \frac{\theta}{t} ]
  • ( \omega ): Angular speed
  • ( \theta ): Central angle in radians
  • ( t ): Time

Example Calculation

• Problem: Object travels around a circle with:
  • Radius: 5 cm
  • Central angle: ( \frac{1}{3} ) radians in 20 seconds
• Angular Speed:
  • ( \omega = \frac{1/3}{20} = \frac{1}{60} ) radians/second, or 0.02 radians/second
• Linear Speed:
  1. Calculate arc length ( s ):
     • ( s = \theta \times r = \frac{1}{3} \times 5 = \frac{5}{3} ) cm
  2. Calculate linear speed ( v ):
     • ( v = \frac{5/3}{20} = \frac{1}{12} ) cm/second, or 0.08 cm/second

Important Notes

• Ensure units are consistent (e.g., radians for angles).
• Check instructions for exact answers or decimal approximations.