Calculating Arc Length from Angular Displacement

The video tutorial explores the concept of angular displacement and its connection to arc length and distance traveled. It uses examples involving a tennis ball tethered to a nail, rotating around a circular path. The tutorial explains how to calculate the distance traveled by the ball using the circumference of the circle and introduces a formula for arc length based on angular displacement and radius. Through various examples, the video demonstrates the application of this formula, emphasizing the relationship between angular displacement and the proportion of the circle's circumference covered.

What is angular displacement in the context of the tennis ball example?

• The length of the string

• The distance the ball travels

• The final position of the ball

• The change in the ball's angle as it moves around the nail

What is the significance of the positive counterclockwise direction mentioned in the video?

• It signifies the ball's displacement is zero

• It is used to calculate the radius of the circle

• It indicates the direction of angular displacement

• It is irrelevant to the calculation of arc length

How is the distance traveled by the ball calculated?

• By determining the ball's final position

• By calculating the circumference of the circle

• By measuring the length of the string

• By multiplying the radius by pi

What is the displacement of the ball in the initial example?

• Six meters

• 12 pi meters

• Two pi radians

• Zero

How does the concept of displacement differ from distance traveled?

• Distance traveled is always zero

• Displacement is always greater than distance traveled

• Displacement can only be measured in radians

• Displacement is a vector quantity, while distance is scalar

In the second scenario, what is the angular displacement when the ball is rotated clockwise?

• Zero

• Two pi radians

• Pi over two radians

• Negative pi over two radians

How is the arc length calculated in the second scenario?

• By multiplying the radius by pi over two

• By taking the absolute value of angular displacement times the radius

• By measuring the length of the string

• By dividing the circumference by four

What role does the radius play in calculating the arc length?

• It is multiplied by the angular displacement to find arc length

• It is irrelevant to the calculation

• It determines the ball's displacement

• It is subtracted from the angular displacement

What does the final example demonstrate about the relationship between angular displacement and arc length?

• That arc length is independent of angular displacement

• That arc length can only be calculated with a radius of five meters

• That angular displacement is not related to the radius of the circle

• That the formula for arc length holds true regardless of string length or angle

What is the arc length in the final example?

• Five pi over six meters

• Pi over six radians

• Five meters

• Pi over three radians