An Atwood Machine has a pulley of radius Rpulley = 0.25m, and
mass Mpulley = 6.0 kg with two blocks with masses m1 = 5.0 kg,
and m2 = 10.0 kg. Treat the pulley as a uniform disk.
a) Find the acceleration of each block.
b) Find the angular acceleration of the pulley.
c) Find the tension in each rope T1 and T2.
To find the answers to these questions, we need to analyze the forces acting on each block and the pulley. We'll start by calculating the acceleration of each block.
a) Acceleration of each block:
1. Determine the net force acting on each block.
For m1: The force on m1 is T1 (tension in the rope) in the upward direction and T2 (tension in the other rope) in the downward direction. The force of gravity acts on m1 in the downward direction.
F_net,m1 = T1 - T2 - m1 * g (where g is the acceleration due to gravity)
For m2: The force on m2 is T2 (tension in the rope) in the upward direction and m2 * g (force of gravity) in the downward direction.
F_net,m2 = T2 - m2 * g
2. Apply Newton's second law (F = m * a) to each block to relate the net force to acceleration.
For m1: F_net,m1 = m1 * a1
For m2: F_net,m2 = m2 * a2
3. Set up and solve a system of equations to find the unknown acceleration(s).
From step 1:
T1 - T2 - m1 * g = m1 * a1
T2 - m2 * g = m2 * a2
Solving this system of equations will give us the accelerations of each block, a1 and a2.
b) Angular acceleration of the pulley:
1. Determine the net torque acting on the pulley.
The net torque on the pulley is caused by the tension in the ropes. One rope pulls clockwise (due to m2's weight) and the other counter-clockwise (due to m1's weight).
Net torque = torque_cw - torque_ccw
Torque_cw = T2 * Rpulley (clockwise torque)
Torque_ccw = T1 * Rpulley (counter-clockwise torque)
2. Apply Newton's second law for rotational motion (τ = Iα) to the pulley, where τ is the net torque, I is the moment of inertia of the pulley, and α is the angular acceleration.
I = (1/2) * Mpulley * Rpulley^2 (moment of inertia for a disk)
Substituting the given values into the equation, we can solve for α.
c) Tension in each rope:
1. Use the net forces we found earlier to calculate the tension in each rope.
For T1, use the equation: T1 = T2 + m1 * g
For T2, use the equation: T2 = (m2 * g - m1 * g) / 2
By following these steps, you should be able to find the desired values for the Atwood Machine system.