Question

The two blocks, m₁ = 2.6 kg and m₂ = 4.2, in the figure below are connected by a massless rope that passes over a pulley. The pulley is 12 cm in diameter and has a mass of 2.0 kg. As the pulley turns, friction at the axle exerts a torque of magnitude 0.55 N · m. If the blocks are released from rest, how long does it take the 4.2 kg block to reach the floor from a height of h = 1.0 m? (Note: If your random numbers do not create movement between the masses enter 0 for your answer.)

Answer 1

The weight of the 4.9 kg block produces a torque on the pulley.
Torque #1 = 4.2 * 9.8 * 0.06  

The weight of the 3.2 kg block produces a torque on the pulley, in the opposite direction of Torque #1.
Torque #2 = -2.6* 9.8 * 0.06

Since Torque #1 is greater than Torque #2, the pulley will rotate in the direction of Torque #2. The torque caused by the friction force must be in the opposite direction of the rotation of the pulley.  
Torque of friction = -0.55

Net torque = Torque #1 + Torque #2 + Torque of friction
Net torque = (4.9 * 9.8 * 0.06) + (-2.6 * 9.8 * 0.06) + -0.55
Net torque = 0.3908 N * m

The net torque will cause the pulley to accelerate as it rotates.
Net torque = Moment of Inertia * angular acceleration

If the pulley is similar to a solid circular disc, the moment of inertia = ½ * mass * radius^2
Moment of inertia = ½ * 2.0 * 0.06^2 = 0.0036

The angular acceleration = Net torque ÷ Moment of Inertia
The angular acceleration = 0.3908 ÷ 0.0036 = 108.55 rad/s

As the pulley rotates the rope moves over the pulley and the 4.2 kg block moves down 1 meter. Since the pulley has angular acceleration, the block is accelerating as it moves down 1 meter.
Linear acceleration = angular acceleration * radius = 108.55 * 0.06 = 6.513 m/s


If the pulley is mass-less and has no friction, what is the acceleration of the 4.2 block.
Net force = 4.2 * 9.8 – 2.6 * 9.8 = 15.68 N
Total mass = 4.2+2.6= 6.8 kg
Acceleration = net force ÷ total mass = 15.68 ÷ 6.8 = 2.3 m/s^2
The linear acceleration must be less than 2.3 m/s^2, because there is friction and the pulley has mass and moment of intertia!


The net torque is not only causing the pulley to accelerate as it rotates. The net torque is also causing the blocks to accelerate.

Total moment of inertia = moment of inertia of blocks and pulley
Moment of inertia of block = mass * (perpendicular distance)^2  

Total moment of inertia = 4.2* 0.06^2 + 2.6* 0.06^2 + 0.0036 = 0.02808
angular acceleration = 0.3908 ÷ 0.02808 = 13.91
Linear acceleration = angular acceleration * radius = 13.91 * 0.06

Linear acceleration = 0.835 m/s^2
This is less than 2.3m/s^2!

The 4.2kg block accelerates at 0.835 m/s^2 as it moves 1 meter down.

Distance = ½ * a * t^2
1 = ½ * 0.835 * t^2
Solve for time
Time = 2.395 seconds