A cylindrical 7.25 kg pulley with a radius of 0.864 m is used to lower a 7.13 kg bucket into a well. The bucket starts from rest and falls for 4.51 s.
What is the linear acceleration of the
falling bucket? The acceleration of gravity is 9.81 m/s^2.
23.4
To find the linear acceleration of the falling bucket, we can apply Newton's second law of motion, which states that the net force acting on an object is equal to its mass multiplied by its acceleration (F = ma).
In this case, the net force acting on the bucket is the force of gravity pulling it downward. The gravitational force can be calculated by multiplying the mass of the bucket (m) by the acceleration due to gravity (g).
The formula for the gravitational force is given by F = mg, where m is the mass of the bucket and g is the acceleration due to gravity. We can substitute the values:
F = (7.13 kg) * (9.81 m/s^2)
F = 69.8973 N
Now, let's calculate the moment of inertia of the pulley. The moment of inertia for a solid cylinder rotating about its central axis is given by the formula I = (1/2) * m * r^2, where m is the mass of the cylinder and r is its radius.
For the given pulley:
m = 7.25 kg
r = 0.864 m
I = (1/2) * (7.25 kg) * (0.864 m)^2
I = 2.8479 kg * m^2
Next, we need to calculate the torque (τ) on the pulley. Since the pulley is free to rotate and there is no external torque applied, the torque is equal to the moment of inertia multiplied by the angular acceleration (τ = I * α).
Solving for α, we find:
τ = I * α
α = τ / I
The torque (τ) acting on the pulley is equal to the force (F) applied by the falling bucket multiplied by the radius (r) of the pulley (τ = F * r). Substituting the values:
τ = (69.8973 N) * (0.864 m)
τ = 60.3165 N * m
Now we can calculate the angular acceleration (α):
α = (60.3165 N * m) / (2.8479 kg * m^2)
α = 21.1604 rad/s^2
Finally, we need to find the linear acceleration (a) of the falling bucket. The linear acceleration is related to the angular acceleration by the equation:
a = α * r
Substituting the values:
a = (21.1604 rad/s^2) * (0.864 m)
a = 18.2484 m/s^2
Therefore, the linear acceleration of the falling bucket is 18.2484 m/s^2.