A belt drives two flywheels whose diameters are 30cm and 45cm, respectively. If the smaller wheel turns through 310 rev/min, find the speed of the belt in meters per second and the regular velocity of the larger wheel in rev/min.
long way:
perimeter of small wheel = 2π(30) cm = 60π cm
so at 310 rotations/min, it has gone 310(60π) = 18600π cm
perimeter of larger wheel = 2π(45) = 90π cm
velocity of larger wheel = 18600π/90π = 206 2/3 rev/min
short way:
by proportion,
30/45 = R/310
R = 310(30)/45 = 206 2/3 rev/min
To solve this problem, we need to use the concept of the ratio of speeds of the belt and the flywheels.
Let's first find the speed of the belt in meters per second.
Given:
Diameter of smaller flywheel (d1) = 30 cm
Diameter of larger flywheel (d2) = 45 cm
The formula to find the speed of the belt (v) is:
v = (π * d1 * N1) / (60 * 100)
where N1 is the speed of the smaller flywheel in revolutions per minute (rev/min).
Substituting the values:
v = (π * 30 * 310) / (60 * 100)
v = (3.14 * 30 * 310) / (60 * 100)
v = (9426 / 60) cm/s
v ≈ 157 cm/s
v ≈ 1.57 m/s
So, the speed of the belt is approximately 1.57 meters per second.
Next, let's find the regular velocity of the larger flywheel in revolutions per minute (rev/min).
The formula to find the regular velocity (N2) is:
N2 = N1 * (d1 / d2)
where N1 is the speed of the smaller flywheel in revolutions per minute (rev/min),
d1 is the diameter of the smaller flywheel, and
d2 is the diameter of the larger flywheel.
Substituting the values:
N2 = 310 * (30 / 45)
N2 = 310 * (2/3)
N2 = 206.666... rev/min
N2 ≈ 207 rev/min
So, the regular velocity of the larger flywheel is approximately 207 revolutions per minute.
To find the speed of the belt in meters per second, we need to know the relationship between the linear speed of the belt and the angular speed of the smaller flywheel.
The linear speed of a point on the edge of a rotating object can be calculated using the formula:
linear speed = radius * angular speed
Since the smaller flywheel has a diameter of 30cm, its radius is half of that, which is 15cm or 0.15m.
Now, we know the angular speed of the smaller flywheel is 310 rev/min. To convert rev/min to radians/second, we need to multiply it by 2π (because there are 2π radians in one revolution) and divide by 60 (since there are 60 seconds in a minute).
angular speed in radians/second = (310 rev/min) * (2π radians/rev) / (60 sec/min)
Plug in the numbers and calculate:
angular speed in radians/second = (310 * 2π) / 60 ≈ 32.54 radians/second
Now, we can calculate the linear speed of the belt:
linear speed = 0.15m * 32.54 radians/second ≈ 4.88 m/s
So, the speed of the belt is approximately 4.88 meters per second.
To find the regular velocity of the larger wheel in rev/min, we can use the fact that the linear speed of the belt is the same for both flywheels.
We already know the linear speed of the belt is 4.88 m/s.
Let's calculate the radius of the larger flywheel. Its diameter is 45cm, so the radius is half of that, which is 22.5cm or 0.225m.
Using the formula linear speed = radius * angular speed, we can solve for the angular speed of the larger flywheel:
4.88 m/s = 0.225m * angular speed of larger flywheel
angular speed of larger flywheel = 4.88 m/s / 0.225m ≈ 21.7 radians/second
To convert this angular speed to rev/min, we can multiply by 60 and divide by 2π:
regular velocity of larger flywheel in rev/min = (21.7 radians/second) * (60 min/rev) / (2π radians) ≈ 327 rev/min
So, the regular velocity of the larger wheel is approximately 327 rev/min.