two pulleys of a car engine radius of 8 cm and 5 cm respectively are connected by a belt. The larger pulley rotates 80 times per minute. Find the angular velocity of each pulley in radians per minute.
Since the radii are in the ratio of 5/8, and they both move the same amount of belt, their speed is in the ratio 8/5.
So, what is 80 * 8/5 ?
128
To find the angular velocity of each pulley in radians per minute, we need to use the formula:
Angular velocity (in radians per minute) = Number of revolutions per minute * 2π
Let's start with the larger pulley, which has a radius of 8 cm and rotates 80 times per minute:
1. First, we need to find the circumference of the larger pulley using the formula: Circumference = 2π * Radius
Circumference = 2π * 8 cm = 16π cm
2. To find the distance traveled in one revolution (360 degrees) by the larger pulley, we use the formula: Distance = Circumference = 16π cm
3. Since the larger pulley rotates 80 times per minute, the total distance traveled in one minute by the larger pulley is:
Total distance = Distance * Number of revolutions = 16π cm * 80 = 1280π cm
4. To convert this distance to radians, we need to use the formula: 1 revolution = 2π radians
Total distance in radians = Total distance / 2π = 1280π cm / 2π = 640 radians
Therefore, the angular velocity of the larger pulley is 640 radians per minute.
Now let's find the angular velocity of the smaller pulley, which has a radius of 5 cm:
1. Using the same method, we find that the total distance traveled by the smaller pulley in one minute is:
Total distance = Circumference * Number of revolutions = 2π * 5 cm * 80 = 800π cm
2. Converting this distance to radians:
Total distance in radians = Total distance / 2π = 800π cm / 2π = 400 radians
Therefore, the angular velocity of the smaller pulley is 400 radians per minute.
In summary:
- The angular velocity of the larger pulley is 640 radians per minute.
- The angular velocity of the smaller pulley is 400 radians per minute.