Angular Acceleration, 2 Pulleys on common axle

In summary, the problem involves finding the angular acceleration of a system consisting of two pulley wheels with different radii and attached masses. Using the equations ΣΤ = Iα and ΣΤ = Τ1 + Τ2, the torques for each mass and radius were calculated. Plugging in the given values and solving for α, the resulting value was 50.90060606 rad/s², which did not match the expected answer. Further checking of the units showed that they were correct, so the error must lie in the calculations.
  • #1
Xlorep
2
0
Note: This is similar to 2 previous questions I've seen here (https://www.physicsforums.com/showthread.php?t=261193"), and while this is my first post, I've lurked/found this site to really be helpful in getting me to be able to figure out how to solve problems I have with my physics homework.

Homework Statement



Two pulley wheels, one of radius 0.3 m and the other of radius 0.82 m, are mounted rigidly on a common axle. The rotational intertia of the two pulleys, which are clamped together, is 3.3 kg·m². A mass of 44 kg mass is attached on the left and a mass of 37 kg mass on the right, as shown.

attachment.php?attachmentid=15717&d=1222991526.jpg


Find the angular acceleration of the system. Take the clockwise direction to be positive. The acceleration of gravity is 9.8 m/s². Answer in units of rad/s².

Homework Equations



ΣΤ = Iα

ΣΤ = Τ1 + Τ2

The Attempt at a Solution



So, the process I followed was this:

I found the torques for each mass & radius:

Τ1 = -(m1)g(R1)sin(90) --> (negative because going CCW)

Τ2 = (m2)g(R2)sin(90) --> (positive because going CW)

Therefore: ΣΤ = (m2)g(R2) - (m1)g(R1) = g(m2R2 - m1R1)

Which means: g(m2R2 - m1R1) = Iα

α = g(m2R2 - m1R1)/I

Checking the units, I should get rad/s²:

m/s²·(kg·m)/kg·m²

Which is: (m²·kg)/(s²·kg·m²)

Results in: rad/s²

So my units check out.

So, with the following variables (I'm hoping I got the left/right values correct):

  • g = 9.8 m/s²
  • m1 = 44 kg
  • R1 = 0.3 m
  • m2 = 37 kg
  • R2 = 0.82 m
  • I = 3.3 kg·m²

Plugging these in, I get:

α = [9.8]([37][0.82] - [44][0.3])/[3.3]

α = (9.8)(30.34 - 13.2)/3.3

α = (9.8)(17.14)/3.3

α = (167.972)/3.3

α = (167.972)/3.3

α = 50.90060606 rad/s²

Which, apparently, is wrong. So I'm at a loss. What did I do wrong? Or rather, what would be a good hint for me to figure out where I went wrong?

(and yes, I know it seems strange for an α > g... that doesn't seem to logically follow for me)
 
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  • #2
Welcome to PF.

Did you enter just 50.9?
 
  • #3
LowlyPion said:
Welcome to PF.

Did you enter just 50.9?

No, but the homework system my teacher uses is very annoying in needing an incredible number of sig figs. So I always have to just leave everything my calculator prints out to have success.

Update: Yes. It didn't help. I'm apparently missing something in my calculations.
 
Last edited:

1. What is angular acceleration?

Angular acceleration is the rate at which the angular velocity of an object changes over time. It is measured in radians per second squared (rad/s²) or degrees per second squared (deg/s²).

2. How is angular acceleration different from linear acceleration?

Angular acceleration is a measure of how quickly an object's angular velocity changes, while linear acceleration is a measure of how quickly an object's linear velocity changes. Angular acceleration involves rotational motion, while linear acceleration involves straight-line motion.

3. What is the formula for calculating angular acceleration?

The formula for angular acceleration is α = (ω₂ - ω₁) / t, where α is the angular acceleration, ω₂ is the final angular velocity, ω₁ is the initial angular velocity, and t is the time interval.

4. How does the presence of two pulleys on a common axle affect the calculation of angular acceleration?

Having two pulleys on a common axle does not affect the calculation of angular acceleration. The same formula can be used, but the angular velocities of both pulleys must be taken into account.

5. What factors can affect the angular acceleration of a system with two pulleys on a common axle?

The angular acceleration of a system with two pulleys on a common axle can be affected by the mass and radius of the pulleys, the tension in the connecting rope or belt, and any external forces acting on the system. Friction between the pulleys and the axle can also affect the angular acceleration.

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