Two pulleys have radii 20cm and 6cm,respectively. The smaller pulley rotates 15 times in 12 seconds. Find the angular speed of each pulley in rad/sec
15rev/12sec = 5/4 rev/sec = 5π/2 rad/s
That's the speed of the smaller pulley
Assuming they are both attached to the same belt, the larger pulley's rotational speed is 6/20 the speed of the smaller one.
Smaller Pulley:
V=15rev/12s * 6.28rad/rev = 7.85 rad/s
Larger Pulley:
V = (6cm/20cm) * 7.85 = 2.36 rad/s.
To find the angular speed of each pulley, we need to determine how many radians the pulleys rotate in one second.
- We can start by finding the angular speed of the smaller pulley.
- The larger pulley has a radius of 20cm, so its circumference (C1) is given by C1 = 2π * 20 cm.
- The smaller pulley has a radius of 6cm, so its circumference (C2) is given by C2 = 2π * 6 cm.
- We know that the smaller pulley rotates 15 times in 12 seconds, which means it completes 15 revolutions in 12 seconds.
- The angular speed of the smaller pulley is then given by the formula ω1 = (2π * 15) / 12 rad/sec.
To find the angular speed of the larger pulley, we can use the fact that the ratio of the angular speeds of two pulleys connected with a belt is equal to the ratio of their radii.
- Let ω2 be the angular speed of the larger pulley.
- Since the ratio of the radii is (20/6), the ratio of the angular speeds is also (20/6).
- Therefore, we have ω2 / ω1 = (20/6).
- Rearranging the equation, we get ω2 = (20/6) * ω1.
Now, substituting the value of ω1 we found earlier, we can calculate ω2.
- ω2 = (20/6) * [(2π * 15) / 12] rad/sec.
Simplifying the equation:
- ω2 = (20/6) * (π * 15/6) rad/sec.
Calculating the value:
- ω2 = (500π/36) rad/sec.
So, the angular speed of the smaller pulley is ω1 = (2π * 15) / 12 rad/sec, and the angular speed of the larger pulley is ω2 = (500π/36) rad/sec.
To find the angular speed of each pulley in rad/sec, we need to first understand the relationship between angular speed, linear speed, and the radius of a pulley.
Angular speed (ω) is the rate at which an object rotates or revolves, usually measured in radians per second (rad/sec). Linear speed (v) is the distance traveled per unit of time, usually measured in meters per second (m/sec). The relationship between these two is given by:
v = ω * r
where r is the radius of the pulley.
Given that the smaller pulley rotates 15 times in 12 seconds, we can calculate its angular speed. The number of revolutions can be converted into radians by multiplying by 2π (since 1 revolution = 2π radians). So, the angular distance covered by the smaller pulley is:
angular distance = 15 * 2π radians
Next, we need to find the time taken for these 15 revolutions. Since 12 seconds is the total time, the time for 15 revolutions is:
time = 12 seconds
Dividing the angular distance by the time gives us the angular speed:
angular speed = angular distance / time
Now, substitute the given values:
angular speed = (15 * 2π radians) / 12 seconds
Simplifying, we get:
angular speed = (15 * π) / 6 radians per second
Thus, the angular speed of the smaller pulley is (15 * π) / 6 rad/sec.
To find the angular speed of the larger pulley, we can use the concept of the ratio of radii. The larger pulley has a radius of 20 cm, while the smaller pulley has a radius of 6 cm. We can determine the ratio of their radii as:
ratio of radii = radius of larger pulley / radius of smaller pulley
= 20 cm / 6 cm
= 3.33
Now, we know that the angular speed is inversely proportional to the radius. So, we can determine the angular speed of the larger pulley using the formula:
angular speed of larger pulley = angular speed of smaller pulley / ratio of radii
Substituting the values we found earlier:
angular speed of larger pulley = (15 * π) / 6 rad/sec / 3.33
Simplifying, we get:
angular speed of larger pulley ≈ (4.5 * π) / 2 rad/sec
Thus, the angular speed of the larger pulley is approximately (4.5 * π) / 2 rad/sec.