assume a pulley has a radius of 12.96 cm. If it takes 18 seconds for 56 cm of belt to go around the pulley, find the angular velocity in radians per seconds
Circumference = 2(pi)12.96 = 81.43
so 18/56 = x/81.43
x= 18(81.43)/56
26.17 sec for 1 rotation of the wheel
so the angular velocity = 2pi/26.17
or .24 radians/second
Well, I hope this doesn't go over your head, but let's solve this step by step!
First, we can find the circumference of the pulley using its radius. The formula for circumference is C = 2πr. Plugging in the radius of 12.96 cm, we get:
C = 2π(12.96) ≈ 81.415 cm
Next, we need to calculate the angular velocity, which is given by the formula ω = θ/t, where θ is the angle (in radians) covered by the belt and t is the time taken.
Since 56 cm of belt went around the pulley, we can find the angle covered by the belt by utilizing the formula θ = s/r, where s is the arc length and r is the radius. Plugging in the corresponding values, we have:
θ = 56 cm / 12.96 cm ≈ 4.32 radians
Finally, we can calculate the angular velocity ω by dividing the angle covered by the time taken:
ω = θ / t = 4.32 radians / 18 s ≈ 0.24 radians per second
So, the angular velocity of the pulley in radians per second is approximately 0.24 rad/s. Remember, don't get too wrapped up in the calculations!
To find the angular velocity in radians per second, we need to determine the angle swept by the pulley in a given time.
The circumference of the pulley can be calculated using its radius:
Circumference = 2 * π * Radius
= 2 * π * 12.96 cm
In this case, the belt covers a distance of 56 cm in 18 seconds.
Therefore, the angle swept by the pulley is given by the formula:
Angle = Distance / Circumference
Substituting the values:
Angle = 56 cm / (2 * π * 12.96 cm)
Now we can find the angular velocity by dividing the angle by the time:
Angular Velocity = Angle / Time
= (56 cm / (2 * π * 12.96 cm)) / 18 seconds
Simplifying:
Angular Velocity = (56 / (2 * π * 12.96)) / 18 radians per second
Calculating this expression will give you the angular velocity in radians per second.
To find the angular velocity in radians per second, we need to relate the linear velocity of the belt to the angular displacement of the pulley.
The linear velocity of the belt can be calculated using the formula:
Linear velocity = Distance / Time
In this case, the distance is given as 56 cm and the time is given as 18 seconds. So, the linear velocity is:
Linear velocity = 56 cm / 18 s = 3.1111 cm/s
Now, to relate the linear velocity to the angular velocity, we use the formula:
Angular velocity = Linear velocity / Radius
The radius of the pulley is given as 12.96 cm. So, the angular velocity is:
Angular velocity = 3.1111 cm/s / 12.96 cm = 0.2395 radians/s
Therefore, the angular velocity of the pulley is approximately 0.2395 radians per second.