In a charming 19th-century hotel, an old-style elevator is connected to a counterweight by a cable that passes over a rotating disk 1.54m in diameter. The elevator is raised and lowered by turning the disk, and the cable does not slip on the rim of the disk but turns with it. At how many rpm(revolutions per minute) must the disk turn to raise the elevator at 27.9 cm/s?
Well, let's analyze the situation! The disk's circumference is given by C = πd, where d is the diameter of the disk. In this case, the circumference of the disk is approximately 4.84 m.
Now, we know that the elevator moves at a speed of 27.9 cm/s or 0.279 m/s. So, every second, the cable wraps around a length of 0.279 meters. To find out how many revolutions per minute (rpm) are needed, we can divide the speed of the elevator by the length of the disk's circumference:
0.279 m/s ÷ 4.84 m = 0.0577 revolutions per second.
Now, to convert that to rpm, we multiply by 60 (since there are 60 seconds in a minute):
0.0577 revolutions per second × 60 seconds = 3.462 rpm.
So, the disk must turn at approximately 3.462 rpm to raise the elevator at 27.9 cm/s.
To determine the number of rpm (revolutions per minute) at which the disk must turn to raise the elevator at 27.9 cm/s, we can use the following steps:
Step 1: Convert the elevator speed from cm/s to m/s:
27.9 cm/s = 0.279 m/s
Step 2: Calculate the circumference of the rotating disk:
Circumference = π * diameter
Circumference = π * 1.54 m
Step 3: Calculate the distance covered by the cable in one revolution:
Distance covered per revolution = Circumference
Step 4: Calculate the number of revolutions needed to raise the elevator at 0.279 m/s:
Number of revolutions = (Speed of elevator)/(Distance covered per revolution)
Step 5: Convert the number of revolutions per second into revolutions per minute:
Number of revolutions per minute = Number of revolutions per second * 60
Let's go ahead and calculate it:
Step 1: Convert the elevator speed from cm/s to m/s:
0.279 m/s
Step 2: Calculate the circumference of the rotating disk:
Circumference = π * 1.54 m ≈ 4.84 m
Step 3: Calculate the distance covered by the cable in one revolution:
Distance covered per revolution = 4.84 m
Step 4: Calculate the number of revolutions needed to raise the elevator at 0.279 m/s:
Number of revolutions = (0.279 m/s) / (4.84 m) ≈ 0.0576 revolutions
Step 5: Convert the number of revolutions per second into revolutions per minute:
Number of revolutions per minute ≈ 0.0576 revolutions * 60 ≈ 3.46 rpm
Therefore, the disk must turn at approximately 3.46 rpm to raise the elevator at 27.9 cm/s.
To calculate the required RPM (revolutions per minute) of the rotating disk in order to raise the elevator at a given speed, we need to use the relationship between the linear speed of the elevator and the rotational speed of the disk.
First, let's determine the circumference of the disk. The circumference is given by the formula:
C = π * d
where C is the circumference and d is the diameter. Plugging in the given diameter of 1.54m, we get:
C = π * 1.54m
C ≈ 4.84m (rounded to two decimal places)
Next, we need to convert the desired linear speed of the elevator from centimeters per second to meters per minute. Since there are 60 seconds in a minute, the conversion factor is 60:
linear speed (m/min) = linear speed (cm/s) * 60
Plugging in the given linear speed of 27.9 cm/s, we get:
linear speed (m/min) = 27.9 cm/s * 60
linear speed (m/min) = 1674 m/min
Now, we can calculate the RPM (revolutions per minute) by dividing the linear speed (m/min) by the circumference (m) of the disk:
RPM = linear speed (m/min) / C
Plugging in the values, we get:
RPM = 1674 m/min / 4.84 m
RPM ≈ 346.28 rpm (rounded to two decimal places)
Therefore, the disk must turn at approximately 346.28 RPM to raise the elevator at a speed of 27.9 cm/s.