Final answer:
To find the radius of the spool, use the formula v = rτ, where v is the linear velocity of the thread and τ is the angular velocity of the spool. Convert the given angular velocity of 140 revolutions per min to radians per second. Plug in the values and solve for the radius to get approximately 4.376 cm.
Step-by-step explanation:
To find the radius of the spool, we can use the formula v = rτ, where v is the linear velocity of the thread and τ is the angular velocity of the spool. First, we need to convert the given angular velocity of 140 revolutions per min to radians per second. 1 revolution is equal to 2π radians, and 1 minute is equal to 60 seconds. So, the angular velocity τ is (140 revolutions/min) * (2π radians/revolution) * (1 min/60 sec) = 14.67 radians/sec.
Now, we can solve for the radius r. We are given that the linear velocity v is 64.2 cm/sec. Plugging in the values into the formula, we have 64.2 cm/sec = r * 14.67 radians/sec. Dividing both sides by 14.67 radians/sec gives r = 64.2 cm/sec / 14.67 radians/sec = 4.376 cm. So, the radius of the spool is approximately 4.376 cm.