the normal acceleration of a particle on the rim of a pulley 10 ft in diameter is constant @ 8000 ft/s^2. determine the speed of the pulley in rpm
Why did the pulley go to the doctor?
Because it needed a "rpm-edy"!
Now, let's calculate the speed of the pulley in rpm.
The normal acceleration of a particle on the rim of a pulley can be related to the linear velocity using the formula:
a_normal = (v^2) / r
where v is the linear velocity and r is the radius of the pulley. Since the pulley has a diameter of 10 ft, the radius is half that, which is 5 ft.
We are given that the normal acceleration is 8000 ft/s^2, so we can rearrange the formula to solve for the velocity:
v = √(a_normal * r)
v = √(8000 ft/s^2 * 5 ft)
v ≈ √(40000 ft^2/s^2)
v ≈ 200 ft/s
Now, we need to convert the velocity from feet per second to revolutions per minute (rpm). Since 1 revolution is equal to the circumference of the pulley, we can calculate the number of revolutions per minute using the formula:
rpm = (v / (2π)) * (60 sec/min)
rpm = (200 ft/s / (2π)) * (60 sec/min)
rpm ≈ (31.83) * (60 sec/min)
rpm ≈ 1909.8
So, the speed of the pulley is approximately 1909.8 rpm.
To determine the speed of the pulley in RPM (revolutions per minute), we need to use the formula:
Speed = (2 * π * r * N) / T
Where:
- Speed is the linear speed of the pulley (in ft/s)
- π is a mathematical constant equal to approximately 3.14159
- r is the radius of the pulley (in ft)
- N is the number of revolutions the pulley makes per minute
- T is the time it takes for one revolution (in seconds)
Given:
- Diameter (d) of the pulley = 10 ft
- Normal acceleration (a) of the particle = 8000 ft/s^2
First, let's calculate the radius (r) of the pulley:
r = d / 2
r = 10 ft / 2
r = 5 ft
Now, let's calculate the time it takes for one revolution (T). Since the pulley's speed is constant, we can use the equation:
a = (4π^2 * r) / T^2
Rearranging the equation, we get:
T^2 = (4π^2 * r) / a
T^2 = (4 * 3.14159^2 * 5) / 8000
T^2 = 0.24664
Taking the square root of both sides, we find:
T ≈ 0.49662 s
Now, we can calculate the number of revolutions per minute (N) using the formula:
N = 60 / T
N = 60 / 0.49662
N ≈ 120.7698
Finally, let's calculate the speed of the pulley in RPM:
Speed = (2 * π * r * N) / T
Speed = (2 * 3.14159 * 5 * 120.7698) / 0.49662
Speed ≈ 3820.091 ft/min
Therefore, the speed of the pulley is approximately 3820.091 RPM.
To determine the speed of the pulley in RPM (revolutions per minute), we need to use the formula that relates acceleration, speed, and radius:
acceleration = (speed)^2 / radius
We are given the normal acceleration of the particle on the rim of the pulley, which is constant at 8000 ft/s^2, and the diameter of the pulley, which we can use to find the radius.
The diameter of the pulley is given as 10 ft. To find the radius, we divide the diameter by 2:
radius = diameter / 2 = 10 ft / 2 = 5 ft
Now we can rearrange the formula to solve for speed:
speed = sqrt(acceleration * radius)
Plugging in the values:
speed = sqrt(8000 ft/s^2 * 5 ft)
= sqrt(40000 ft^2/s^2)
= 200 ft/s
Now we can convert the speed from feet per second to revolutions per minute.
First, let's convert it to feet per minute:
speed_ft_per_min = speed * 60
= 200 ft/s * 60
= 12000 ft/min
Since the circumference of a circle is equal to 2π times the radius, we can find the distance traveled by the particle on the rim in one revolution:
distance_per_revolution = 2π * radius = 2π * 5 ft
Now we can calculate the number of revolutions per minute:
RPM = speed_ft_per_min / distance_per_revolution
= 12000 ft/min / (2π * 5 ft/rev)
= 12000 ft/min / (10π ft/rev)
= 1200 / π rev/min
So, the speed of the pulley is approximately 1200 / π rpm.
r = 5 ft
a = v^2/r = v^2/5 = 8000
v^2 = 40,000
v = 200 ft/s
v = w r = 2 pi f r
200 = 2 pi f (5)
2 pi f = 40 radians/sec
40 rad/s (1 rev/2 pi rad)(60 s/min)
= 392 revs/min