A flywheel turning at 1200 rev/min (125.7 rad/s) constant angular velocity has a radius of 2.5 cm. As it turns, a string is to be wound onto its rim. How long is the string if it is wound up in 10 seconds?
It says the answer is 31.4m, but I don't know the equations to get this answer.
10 seconds is 1/6 minute, so that would be 200 revolutions
The circumference of the rim is 2pi*r = 5pi cm
200*5pi = 1000pi = 3141.6 cm = 31.4m
To find the length of the string wound onto the flywheel, you need to use the formula for the length of a circular arc.
The formula for the length of a circular arc is given by:
Length = radius × angle in radians
In this case, the angular velocity of the flywheel is given as 125.7 rad/s, and it takes 10 seconds to wind the string. Therefore, you need to find the angle through which the flywheel has turned in 10 seconds.
To do this, multiply the angular velocity by the time:
Angle = angular velocity × time
Angle = 125.7 rad/s × 10 s = 1257 rad
Now, we can calculate the length of the string using the radius and the angle:
Length = radius × angle
Length = 2.5 cm × 1257 rad
Remember to convert the radius from centimeters to meters:
Length = 0.025 m × 1257 rad
Length = 31.4 meters
Therefore, the length of the string wound onto the flywheel in 10 seconds is 31.4 meters.
To find the length of the string wound onto the flywheel's rim, we can use the formula for the circumference of a circle. The circumference of a circle is given by the equation C = 2πr, where C is the circumference and r is the radius.
In this case, the radius of the flywheel is given as 2.5 cm, which we need to convert to meters (since the final answer is in meters). There are 100 centimeters in 1 meter, so the radius in meters is 2.5 cm / 100 = 0.025 m.
Now, let's calculate the total number of revolutions made by the flywheel in 10 seconds. Given that the angular velocity is 125.7 rad/s, we can use the formula θ = ωt, where θ is the angular displacement, ω is the angular velocity, and t is the time. In this case, we have θ = 125.7 rad/s * 10 s = 1257 rad.
Since we know that one revolution is equal to 2π radians, we can find the total number of revolutions by dividing the angular displacement by 2π: 1257 rad / (2π rad/rev) ≈ 200.15 rev.
Finally, we can calculate the length of the string wound onto the flywheel by multiplying the total number of revolutions by the circumference of the flywheel. Using the formula C = 2πr, the length of the string is 200.15 rev * (2π * 0.025 m) = 31.42 m, which can be rounded to 31.4 m.
Therefore, the length of the string wound onto the flywheel's rim is approximately 31.4 m.