A 1.40 kg mass is attached to a light cord that is wrapped around a pulley of radius 3.50 cm, which turns with negligible friction. The mass falls at a constant acceleration of 2.60 m/s2. Find the moment of inertia of the pulley.
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To find the moment of inertia of the pulley, we need to use the equation:
Moment of Inertia = (mass * radius^2) / 2
In this case, the mass of the pulley is not given. However, we can use the given information to calculate the mass using Newton's second law.
The force causing the acceleration of the mass can be determined using Newton's second law:
Force = mass * acceleration
In this case, the force is caused by the tension in the cord. The tension in the cord, T, is related to the mass by the equation:
T = mass * acceleration + mg
where g is the acceleration due to gravity.
We can rearrange this equation to solve for the mass:
mass = (T - mg) / acceleration
The tension in the cord, T, can be calculated using the following equations:
T = m * acceleration + mg
T = I * angular acceleration / radius
Equating the two expressions for T, we can solve for the moment of inertia, I:
I = (m * acceleration + mg) * radius / angular acceleration
Now we have all the information needed to calculate the moment of inertia of the pulley.
First, calculate the mass of the hanging mass:
mass = (T - mg) / acceleration
mass = (m * acceleration + mg - mg) / acceleration
mass = m
Therefore, the mass of the hanging mass is 1.40 kg.
Now, substitute the mass and other given values into the equation for the moment of inertia:
I = (m * acceleration + mg) * radius / angular acceleration
I = (1.40 kg * 2.60 m/s^2 + 1.40 kg * 9.8 m/s^2) * 0.035 m / 2.60 m/s^2
Finally, compute the moment of inertia:
I = (3.64 N + 13.72 N) * 0.035 m / 2.60 m/s^2
I = 0.50 kg·m^2
Therefore, the moment of inertia of the pulley is 0.50 kg·m^2.