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Chapter 5: Problem 2

A soccer ball of diameter \(31 \mathrm{cm}\) rolls without slipping at a linear speed of \(2.8 \mathrm{m} / \mathrm{s} .\) Through how many revolutions has the soccer ball turned as it moves a linear distance of \(18 \mathrm{m} ?\)

Short Answer

Expert verified
Answer: 18 complete revolutions.

Step by step solution

01

Convert diameter to meters and calculate the circumference

To calculate the soccer ball's circumference, first convert the diameter from centimeters to meters. Since there are 100 centimeters in a meter, divide the diameter by 100: Diameter = 31 cm = 31/100 m = 0.31 m. Now, apply the formula for the circumference of a circle, which is given by: Circumference = 2 * pi * radius The radius is half the diameter: Radius = Diameter / 2 = 0.31 m / 2 = 0.155 m Circumference = 2 * pi * 0.155 m = 0.974 m (approximately)
02

Calculate the distance traveled by a point on the ball's surface

We know the linear speed of the soccer ball (2.8 m/s) and the distance it travels (18 m). To find the total distance traveled by a point on the soccer ball's surface, multiply the linear speed by the time taken to travel the distance: Distance traveled = Linear speed * Time We can find the time by dividing the distance traveled by the linear speed: Time = Distance traveled / Linear speed Time = 18 m / 2.8 m/s = 6.43 s (approximately) Now, find the distance traveled by a point on the ball's surface: Distance traveled = Linear speed * Time Distance traveled = 2.8 m/s * 6.43 s = 18 m
03

Determine the number of complete revolutions

To find out how many complete revolutions the soccer ball made, divide the distance traveled by a point on the ball's surface by the soccer ball's circumference: Number of revolutions = Distance traveled / Circumference Number of revolutions = 18 m / 0.974 m = 18.48 (approximately) Since a ball cannot make a non-integer number of revolutions, we round down to the nearest whole number: Number of revolutions = 18 complete revolutions

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