Two masses are suspended from a pulley as shown in the figure . The pulley itself has a mass of 0.30 kg, a radius of 0.20 m, and a constant torque of 0.30 m \cdot N due to the friction between the rotating pulley and its axle.
What is the magnitude of the acceleration of the suspended masses if m_1 = 0.40 kg and m_2 = 0.90 kg ? (Neglect the mass of the string.)
To find the magnitude of the acceleration of the suspended masses, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
In this case, there are two masses suspended from the pulley, so we need to consider the forces acting on each mass separately.
Let's consider mass m1 first. The net force acting on m1 is the tension in the string on the left side of the pulley. This tension is equal to the weight of m1 (m1 * g, where g is the acceleration due to gravity) minus any force due to the friction between the pulley and its axle. So, the net force on m1 is:
F_net1 = m1 * g - Tension(left side)
Now, let's consider mass m2. The net force acting on m2 is the tension in the string on the right side of the pulley. Again, this tension is equal to the weight of m2 (m2 * g) minus any force due to friction. So, the net force on m2 is:
F_net2 = m2 * g - Tension(right side)
Since the pulley is in rotational equilibrium (constant torque), we can write the equation:
Tension(right side) - Tension(left side) = Torque/friction constant
In this case, the torque due to friction is given as 0.30 m * N. Substituting the values into the equation and solving for Tension(right side), we get:
Tension(right side) = Torque/friction constant + Tension(left side)
Now, we can substitute the tensions in the net force equations:
F_net1 = m1 * g - Tension(left side)
F_net2 = m2 * g - (Torque/friction constant + Tension(left side))
Since the masses are connected by a string, the magnitude of their respective accelerations is the same. So, let's call it 'a'. Using Newton's second law, we can write:
F_net1 = m1 * a
F_net2 = m2 * a
Now, we can substitute the above equations for F_net1 and F_net2:
m1 * a = m1 * g - Tension(left side)
m2 * a = m2 * g - (Torque/friction constant + Tension(left side))
Finally, we have two equations with two unknowns (a and Tension(left side)). Solving these equations simultaneously will give us the magnitude of the acceleration of the suspended masses.