Angular Acceleration of a system.

[myr10]

Homework Statement

Two pulley wheels, or respective radii R1 =
0.34 m and R2 = 0.64 m are mounted rigidly
on a common axle and clamped together. The
combined moment of inertia of the two wheels
is I + 1.2 kg*m^2.
Mass m1 = 25 kg is attached to a cord
wrapped around the first wheel, and another
mass m2 = 19 kg is attached to another cord
wrapped around the second wheel.

The acceleration of gravity is 9.8 m/s^2 :
Find the angular acceleration of the system.
Take clockwise direction as positive. Answer
in units of rad/s^2.

Homework Equations

The Attempt at a Solution

im not sure how to go about solving this.

 

[tiny-tim]

Welcome to PF!

Two pulley wheels, or respective radii R1 =
0.34 m and R2 = 0.64 m are mounted rigidly
on a common axle and clamped together. The
combined moment of inertia of the two wheels
is I + 1.2 kg*m^2.
Mass m1 = 25 kg is attached to a cord
wrapped around the first wheel, and another
mass m2 = 19 kg is attached to another cord
wrapped around the second wheel.

Hi myr10! Welcome to PF!

(have a squared: ² and an omega: ω )

(I assume you meant to type I = 1.2 kg*m²)

Hint: what is the torque of the tensions in the two cords?

 

[baba89]

I would approach this problem by first understanding the basic principles of rotational motion. In this system, there are two pulley wheels with different radii that are connected by a common axle. The moment of inertia of the combined system is given, and there are two masses attached to the pulley wheels.

To find the angular acceleration of the system, we can use the equation:

α = τ/I

where α is the angular acceleration, τ is the net torque acting on the system, and I is the moment of inertia. In this case, we can calculate the net torque by considering the forces acting on each mass.

For the first mass, we have a tension force acting downward on the cord, and a weight force acting downward as well. The tension force also creates a torque on the pulley wheel, given by τ = F * R1, where F is the tension force and R1 is the radius of the first pulley wheel. The weight force creates a torque in the opposite direction, given by τ = mg * R1, where m is the mass of the first object and g is the acceleration due to gravity. Therefore, the net torque on the first pulley wheel is:

τ1 = F * R1 - mg * R1 = (F - mg) * R1

Similarly, for the second mass, the net torque is:

τ2 = (F - mg) * R2

Since the two pulley wheels are clamped together, the net torque on the system is equal to the sum of the individual torques:

τ = τ1 + τ2 = [(F - mg) * R1] + [(F - mg) * R2]

We can then plug this into our equation for angular acceleration:

α = [(F - mg) * R1 + (F - mg) * R2] / (I + 1.2)

To find the tension force, we can use Newton's second law:

F = m * a

where m is the mass of the second object and a is its linear acceleration. Since the pulley wheel is rotating at a constant angular velocity, we can relate the linear acceleration to the angular acceleration using the equation:

a = R2 * α

Substituting this into our equation for the tension force, we get:

F = m * R2 * α

Finally, we can substitute this into our equation for angular acceleration to get:

α = [(