an atwood machine is constructed by mounting a well-lubricated pulley (not necessarily a ring/disc), whose radius is 6.00cm on an axle and passing a rope over the pulley. Each end of the rope is attached to a different mass of 3.0 kg and 2.0kg so they both hang vertically. when released from rest, a motion detector measures the descending mass's acceleration as 0.70m/s^2. The mass of the pulley is 4.00kg.
a.) Find the rotational inertia of the pulley.
b) find the kinetic energy of the ENTIRE system when the mass has fallen 3.5 m.
a) To find the rotational inertia of the pulley, we can use the equation:
I = (1/2) * m * r^2
Where:
I = rotational inertia
m = mass of the pulley
r = radius of the pulley
Given:
m = 4.00 kg
r = 6.00 cm = 0.06 m
Substituting the values into the equation, we get:
I = (1/2) * 4.00 kg * (0.06 m)^2
Simplifying the expression, we have:
I = 0.72 kg * m^2
Therefore, the rotational inertia of the pulley is 0.72 kg * m^2.
b) To find the kinetic energy of the entire system when the mass has fallen 3.5 m, we need to consider both the translational kinetic energy of the masses and the rotational kinetic energy of the pulley.
Translational kinetic energy:
The translational kinetic energy of an object is given by the equation:
KEt = (1/2) * m * v^2
Where:
KEt = translational kinetic energy
m = mass of the object
v = velocity of the object
For the descending mass of 3.0 kg, we can calculate the translational kinetic energy as follows:
KEt1 = (1/2) * 3.0 kg * (0.70 m/s)^2
For the ascending mass of 2.0 kg, the velocity is the same but in the opposite direction. So, the translational kinetic energy for the ascending mass is:
KEt2 = (1/2) * 2.0 kg * (0.70 m/s)^2
Rotational kinetic energy:
The rotational kinetic energy of an object is given by the equation:
KEr = (1/2) * I * ω^2
Where:
KEr = rotational kinetic energy
I = rotational inertia
ω = angular velocity
Since the mass is falling vertically, the angular velocity can be calculated using the equation:
ω = v / r
Where:
v = velocity of the descending mass
r = radius of the pulley
First, let's calculate the angular velocity:
ω = 0.70 m/s / 0.06 m = 11.67 rad/s
Now, we can calculate the rotational kinetic energy of the pulley:
KEr = (1/2) * 0.72 kg * m^2 * (11.67 rad/s)^2
Finally, to find the total kinetic energy of the system, we add up the translational kinetic energies and the rotational kinetic energy:
KE_total = KEt1 + KEt2 + KEr
Please substitute the calculated values into the respective equations to find the answer.