A block (mass 2.4 kg) is hanging from a massless cord that is wrapped around a pulley
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(moment of inertia of the pulley = 1.3 x 10 kg·m ), as the drawing shows. Initially the
pulley is prevented from rotating and the block is stationary. Then, the pulley is allowed to rotate as the block falls. The cord does not slip relative to the pulley as the block falls. Assume that the radius of the cord around the pulley remains constant at a value of 0.047 m during the block's descent. Find:
a) the angular acceleration of the pulley and
b) the tension in the cord.
To find the angular acceleration of the pulley, we can use the rotational equivalent of Newton's second law, which states that the torque applied to an object is equal to the moment of inertia of the object multiplied by its angular acceleration. The equation for torque is:
τ = Iα
Where τ is the torque, I is the moment of inertia, and α is the angular acceleration.
In this case, since the pulley is initially prevented from rotating, the net torque on the pulley is zero. But once the block starts to fall, the tension in the cord will exert a torque on the pulley. The torque due to the tension can be calculated as:
τ = r * T
Where r is the radius of the cord around the pulley and T is the tension in the cord.
Since the moment of inertia of the pulley is given as 1.3 x 10 kg·m², we can set up the equation:
r * T = I * α
Substituting the given values, we have:
(0.047 m) * T = (1.3 x 10 kg·m²) * α
Now let's move on to finding the tension in the cord.
The block is hanging, so there is a gravitational force acting on it. This force can be split into two components: one along the cord and one perpendicular to the cord. The component along the cord is equal to the tension in the cord, and the component perpendicular to the cord is equal to the weight of the block.
The weight of the block can be found using the equation:
F = m * g
Where F is the force, m is the mass, and g is the acceleration due to gravity.
In this case, the mass of the block is given as 2.4 kg. Assuming g is approximately 9.8 m/s², we can calculate the weight:
F = (2.4 kg) * (9.8 m/s²)
Now, since the component of the weight along the cord is equal to the tension in the cord, we have:
T = (2.4 kg) * (9.8 m/s²)
Once we have the tension in the cord, we can substitute it back into the previous equation to solve for the angular acceleration of the pulley.