The 2 pulleys in the figure have radii of 15 cm and 8 cm, respectively. The larger pulley rotates 25 times in 36 seconds find the angular velocity of each pulley in radians per second. Find angular velocity of each pulley in radians per second.
2π(15)/(2π(8)) = 1.875 or 15/8
so for each rotation of the large pulley, the smaller rotates 1.875 times or 15/8 times.
angular velocity of large one = 25(2π)/36 radians per second
= 25π/18 rad/s
angular velocity of smaller = (15/8)(25π/18) rad/s
= 125π/48 rad/s
Well, let's calculate the angular velocity of each pulley. To do that, we need to use the formula:
Angular velocity (in radians per second) = (Number of revolutions * 2π) / Time taken (in seconds)
For the larger pulley:
Number of revolutions = 25
Radius = 15 cm = 0.15 meters (converting to meters)
Time taken = 36 seconds
So, the angular velocity of the larger pulley is:
(25 revolutions * 2π) / 36 seconds = 1.38 radians per second
Now let's calculate the angular velocity of the smaller pulley:
Number of revolutions = ?
Radius = 8 cm = 0.08 meters (converting to meters)
Time taken = 36 seconds
Since we know the ratio of the radii of the pulleys (15:8), we can use this ratio to find the number of revolutions of the smaller pulley.
(0.15 meters / 0.08 meters) * 25 revolutions = 46.88 revolutions (rounded to 2 decimal places)
Now we can calculate the angular velocity of the smaller pulley:
(46.88 revolutions * 2π) / 36 seconds = 8.23 radians per second
So, the angular velocity of the larger pulley is approximately 1.38 radians per second, and the angular velocity of the smaller pulley is approximately 8.23 radians per second.
To find the angular velocity of each pulley, we need to determine the number of seconds it takes for each pulley to complete one full rotation.
The larger pulley rotates 25 times in 36 seconds, so the time taken for one full rotation is:
36 seconds ÷ 25 rotations = 1.44 seconds/rotation
The smaller pulley has a radius of 8 cm, while the larger pulley has a radius of 15 cm. We know that the linear velocity at any point on a rotating object is equal to the radius multiplied by the angular velocity. Since both pulleys are rotating at the same angular velocity, the linear velocity at the edge of the larger pulley must be equal to the linear velocity at the edge of the smaller pulley. Therefore, we can set up the following equation:
Linear velocity of larger pulley = Linear velocity of smaller pulley
15 cm × angular velocity of larger pulley = 8 cm × angular velocity of smaller pulley
Simplifying this equation, we have:
angular velocity of larger pulley = (8 cm / 15 cm) × angular velocity of smaller pulley
Now, let's substitute the time taken for one full rotation of the larger pulley into the equation:
angular velocity of larger pulley = (8 cm / 15 cm) × (2π radians/1.44 seconds)
Simplifying this equation, we get:
angular velocity of larger pulley = (8/15) × (2π/1.44) radians/seconds
Evaluate this expression to find the angular velocity of the larger pulley in radians per second.
Similarly, we can find the angular velocity of the smaller pulley by substituting the calculated angular velocity of the larger pulley into the equation:
angular velocity of smaller pulley = (15 cm / 8 cm) × (angular velocity of larger pulley)
Evaluate this expression to find the angular velocity of the smaller pulley in radians per second.
To find the angular velocity of each pulley in radians per second, we need to use the formula:
Angular velocity (in radians per second) = Number of revolutions * 2π / Time (in seconds)
Let's start with the larger pulley.
1. Calculate the number of revolutions made by the larger pulley in 36 seconds:
Number of revolutions = 25
2. Calculate the angular velocity of the larger pulley:
Angular velocity (larger pulley) = 25 * 2π / 36
Angular velocity (larger pulley) = (25 * 2 * 3.14159) / 36
Angular velocity (larger pulley) ≈ 4.3633 radians per second
Now let's move on to the smaller pulley.
3. Determine the ratio of the radii of the pulleys:
Ratio of radii = Radius of larger pulley / Radius of smaller pulley
Ratio of radii = 15 cm / 8 cm
Ratio of radii = 1.875
4. Since the larger pulley rotates 1.875 times faster than the smaller pulley (due to the ratio of the radii), we can calculate the angular velocity of the smaller pulley:
Angular velocity (smaller pulley) = Angular velocity (larger pulley) * Ratio of radii
Angular velocity (smaller pulley) = 4.3633 * 1.875
Angular velocity (smaller pulley) ≈ 8.1925 radians per second
Therefore, the angular velocity of the larger pulley is approximately 4.3633 radians per second, and the angular velocity of the smaller pulley is approximately 8.1925 radians per second.