a belt drives two flywheels whose diameters are in 30 cm and 40 cm, respectively. If the smaller wheel turns through 300 rev/min, what is the speed of the belt in mps and the angular velocity of the larger wheel in rev/min

the angular velocities of the belt-linked wheels

... are inversely proportional to their diameters

so the larger wheel is ... 3/4 * 300 rpm

300 rpm is 5 rps ... belt speed is ... π * .3 m * 5 ... mps

Well, it sounds like this belt has a lot of responsibility! Let's calculate the speed of the belt first.

The speed of the belt can be determined by finding the circumference of the smaller wheel and multiplying it by the number of revolutions per minute:

Circumference of the smaller wheel = π * diameter of the smaller wheel
= π * 30 cm
≈ 94.25 cm

Speed of the belt = Circumference of the smaller wheel * number of revolutions per minute
= 94.25 cm * 300 rev/min
≈ 28,275 cm/min

To convert this speed to meters per second (mps), we'll divide by 100 to convert centimeters to meters, and by 60 to convert minutes to seconds:

Speed of the belt = 28,275 cm/min ÷ (100 cm/m) ÷ (60 m/s)
≈ 4.71 m/s

So, the speed of the belt is approximately 4.71 m/s.

Now, let's calculate the angular velocity of the larger wheel in revolutions per minute.

The angular velocity of a wheel can be determined by using the ratio of the diameters of the two wheels:

Angular velocity of the larger wheel = (diameter of the smaller wheel / diameter of the larger wheel) * angular velocity of the smaller wheel

Angular velocity of the larger wheel = (30 cm / 40 cm) * 300 rev/min
≈ 225 rev/min

Therefore, the angular velocity of the larger wheel is approximately 225 rev/min.

Well, now you know the speed of the belt and the angular velocity of the larger wheel. Belt up and enjoy the ride!

To find the speed of the belt in meters per second (mps), we need to determine the linear velocity of the belt.

The linear velocity (v) of the belt is given by the formula:

v = ω * r

where:
v = linear velocity
ω = angular velocity
r = radius of the wheel

We know that the diameter of the smaller flywheel is 30 cm, so the radius (r1) can be calculated as half of the diameter:

r1 = 30 cm / 2 = 15 cm = 0.15 m

The smaller wheel's angular velocity (ω1) is given as 300 rev/min. To convert it to radians per second (rad/s), we multiply it by 2π:

ω1 = 300 rev/min * 2π rad/rev * (1 min / 60 s) = 300 * π / 30 rad/s
= 10π rad/s

Now, we can calculate the linear velocity (v1) of the belt using the formula:

v1 = ω1 * r1

v1 = (10π rad/s) * (0.15 m)
≈ 4.71 m/s

Therefore, the speed of the belt is approximately 4.71 m/s.

To find the angular velocity of the larger wheel in revolutions per minute (rev/min), we can use the relationship between the angular velocities of the two wheels.

The angular velocity (ω) is related to the linear velocity (v) and the radius (r) by the formula:

v = ω * r

For the larger wheel, we know the linear velocity (v2) of the belt is the same as v1 because they are connected:

v2 = v1 ≈ 4.71 m/s

We need to calculate the radius (r2) of the larger wheel. The diameter of the larger flywheel is given as 40 cm:

r2 = 40 cm / 2 = 20 cm = 0.20 m

Substituting the values of v2 and r2 into the formula, we get:

v2 = ω2 * r2

4.71 m/s = ω2 * 0.20 m

Solving for ω2:

ω2 = 4.71 m/s / 0.20 m
≈ 23.55 rad/s

To convert this to revolutions per minute (rev/min), we divide by 2π and multiply by 60:

ω2 = 23.55 rad/s * (1 rev / 2π rad) * (60 s / 1 min)
≈ 224.46 rev/min

Therefore, the angular velocity of the larger wheel is approximately 224.46 rev/min.

To find the speed of the belt in meters per second (mps) and the angular velocity of the larger wheel in rev/min, we can use the relationship between angular velocity and linear velocity.

Angular velocity (ω) is the rate at which an object rotates, usually measured in radians per second (rad/s) or revolutions per minute (rev/min). It is equal to the ratio of the angle covered (in radians) to the elapsed time.

Linear velocity (v), on the other hand, is the rate at which an object moves along a circular path. It is the product of the angular velocity (ω) and the radius (r) of the circular path.

To calculate the speed of the belt in mps:

1. Start by converting the given angular velocity of the smaller wheel into radians per second. Since 1 revolution is equal to 2π radians, multiply the given value of 300 rev/min by 2π/60 to get the angular velocity in radians per second.

Angular velocity of the smaller wheel = 300 rev/min * (2π radians/1 rev) * (1 min/60 sec) = 10π radians/sec

2. The belt is in contact with both flywheels, so its linear velocity will be the same as the linear velocity of either wheel.

Since the linear velocity (v) is equal to the angular velocity (ω) multiplied by the radius (r), we need to determine the radius of the smaller wheel. The radius is half the diameter. Therefore, r = 30 cm / 2 = 15 cm = 0.15 m.

Linear velocity of the belt = Angular velocity of the smaller wheel * Radius of the smaller wheel
= 10π radians/sec * 0.15 m
= 1.5π m/s

Thus, the speed of the belt is approximately 4.71 m/s (rounded to two decimal places).

To find the angular velocity of the larger wheel in rev/min:

We can use the relationship between the linear velocity (v) of a point on the larger wheel and the angular velocity (ω) of the larger wheel.

Linear velocity of a point on the larger wheel = Angular velocity of the larger wheel * Radius of the larger wheel

We already know the linear velocity of the belt (1.5π m/s) and the radius of the larger wheel (40 cm / 2 = 20 cm = 0.2 m).

Therefore:

Linear velocity of a point on the larger wheel = 1.5π m/s = Angular velocity of the larger wheel * 0.2 m

Solving for the angular velocity of the larger wheel:

Angular velocity of the larger wheel = (1.5π m/s) / (0.2 m)
= 7.5π rev/min

Thus, the angular velocity of the larger wheel is approximately 23.56 rev/min (rounded to two decimal places).

So, the speed of the belt is approximately 4.71 m/s, and the angular velocity of the larger wheel is approximately 23.56 rev/min.