To find the acceleration of the suspended masses, we need first to determine the net torque acting on the system. This can be obtained by considering the torques due to the gravitational forces acting on the masses and the frictional torque.
Let tension on string T1 be acting on m1 and tension T2 acting on m2, and let the angular acceleration of the pulley be α.
For mass m1:
m1 * g - T1 = m1 * a (1)
For mass m2:
T2 - m2 * g = m2 * a (2)
Next, we apply Newton's second law in rotational form, to the pulley.
The torque caused by T1 and T2 is given by:
τ_net = T2 * r - T1 * r = I * α (3)
Here, τ_net is the net torque acting due to tensions T1 and T2 and r is the radius of the pulley, whereas I is the moment of inertia of the pulley.
But the α is also equal to a / r (i.e., acceleration divided by the radius of the pulley).
Substitution α with a/r and the fact that the moment of inertia of a solid cylindrical pulley is given by I = (1/2) * M_pulley * r^2, we get:
τ_net = T2 * r - T1 * r = (1/2) * M_pulley * r * a (4)
Using equations (1) and (2), we can express T1 and T2 in terms of m1 * a and m2 * a, thus:
m1 * g - m1 * a = T1
T2 = m2 * a + m2 * g
Now substitute these expressions for T1 and T2 into equation 4 and obtain an equation in terms of a only:
τ_net = (m2 * a + m2 * g - m1 * g + m1 * a) * r = (1/2) * M_pulley * r * a
Substitute the given values to calculate the acceleration (remember to also include the frictional torque in your calculation):
0.35 N = ((0.7 kg * a + 0.7 kg * 9.8 m/s² - 0.3 kg * 9.8 m/s² + 0.3 kg * a) * 0.145 m) = (1/2) * 0.2 kg * 0.15 m * a
0.35 N = (0.7 * a + 6.86 - 2.94 + 0.3 * a) * 0.15
0.35 N = (1 * a + 3.92) * 0.15
0.35 N = 0.15 * a + 0.588
a = (0.35 - 0.588) / 0.15
a = -1.59 m/s²
Since we are asked for the magnitude of the acceleration, we take the absolute value:
a = 1.59 m/s²
Two masses are suspended from a pulley as shown in the figure below (the Atwood machine). The pulley itself has a mass of 0.20 kg, a radius of 0.15 m, and a constant torque of 0.35 m·N due to the friction between the rotating pulley and its axle. What is the magnitude of the acceleration of the suspended masses if m1 = 0.30 kg and m2 = 0.70 kg? (Neglect the mass of the string.)
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