(34) A 6 FOOT (IN DIAMETER) ROTATES AT HOO REVOLUTIONS PER MIMUTE (a) FIND THE ANGULAR SPEED IN RADIANS PER MINUTE (b) FIND LINEAR SPEED IN FEET PER MINUTE

To solve the problem, we need to find both the angular speed in radians per minute and the linear speed in feet per minute for a circle with a diameter of 6 feet that rotates at 100 revolutions per minute (RPM). ### Step 1: Extract Known Conditions - Diameter of the circle, \( d = 6 \) feet - Radius of the circle, \( r = \frac{d}{2} = \frac{6}{2} = 3 \) feet - Revolutions per minute, \( n = 100 \) RPM ### Step 2: Find Angular Speed in Radians per Minute Angular speed \( \omega \) in radians per minute can be calculated using the formula: \[ \omega = n \times 2\pi \] where \( n \) is the number of revolutions per minute and \( 2\pi \) is the number of radians in one revolution. ### Step 3: Find Linear Speed in Feet per Minute Linear speed \( v \) can be calculated using the formula: \[ v = r \times \omega \] where \( r \) is the radius and \( \omega \) is the angular speed in radians per minute. Now, let's perform the calculations. 1. Calculate the angular speed \( \omega \). 2. Calculate the linear speed \( v \). Let's compute these values. Calculate the value by following steps: - step0: Calculate: \(3\left(100\times 2\pi \right)\) - step1: Remove the parentheses: \(3\times 100\times 2\pi \) - step2: Multiply the terms: \(600\pi \) Calculate or simplify the expression \( 100 * 2 * \pi \). Calculate the value by following steps: - step0: Calculate: \(100\times 2\pi \) - step1: Multiply the terms: \(200\pi \) ### Step 4: Results 1. **Angular Speed**: - The angular speed \( \omega \) in radians per minute is: \[ \omega = 100 \times 2\pi \approx 628.32 \text{ radians per minute} \] 2. **Linear Speed**: - The linear speed \( v \) in feet per minute is: \[ v = 3 \times (100 \times 2\pi) \approx 1884.96 \text{ feet per minute} \] ### Final Answers: - (a) The angular speed is approximately \( 628.32 \) radians per minute. - (b) The linear speed is approximately \( 1884.96 \) feet per minute.