A 3.28-kg bucket is attached to a disk-shaped pulley of radius 0.108 m and mass 0.727 kg.
(a) If the bucket is allowed to fall, what is its linear acceleration?
(b) What is the angular acceleration of the pulley?
(c) How far does the bucket drop in 1.65 s?
T = tension in line
free body of bucket (Positive is down):
mg - T = m a
free body of pulley:
acceleration a = angular acceleration alpha * r
alpha = moment/I
I = (1/2) m r^2 =
moment = T r
so
alpha = 2 T r/ m r^2 = 2 T/m r
a = alpha r = 2 T/m
therefore
mg - T = m a but a = 2 T/m
so
m g - T = 2T
T = m g/3
now go back and get a and alpha etc
this is wrong. periodt.
To find the linear acceleration of the bucket, we need to apply Newton's second law of motion. The formula for linear acceleration is given as:
acceleration = net force / mass
(a) To calculate the net force, we need to consider the forces acting on the bucket. In this case, the only force involved is the weight of the bucket. The weight can be calculated using the equation:
weight = mass * gravity
where mass is the mass of the bucket (3.28 kg) and gravity is the acceleration due to gravity (9.8 m/s^2). Therefore, the weight of the bucket is:
weight = 3.28 kg * 9.8 m/s^2 = 32.144 N
Now, since the pulley system is assumed to be ideal (without friction), the net force on the bucket is equal to the weight. Therefore, the net force is 32.144 N.
Next, we can calculate the linear acceleration using the formula mentioned earlier:
acceleration = net force / mass
acceleration = 32.144 N / 3.28 kg = 9.8 m/s^2
So, the linear acceleration of the bucket is 9.8 m/s^2.
(b) To find the angular acceleration of the pulley, we can use the formula:
angular acceleration = net torque / moment of inertia
The net torque acting on the pulley is equal to the force applied multiplied by the radius of the pulley. In this case, the weight of the bucket (32.144 N) acts tangentially at the radius of the pulley (0.108 m). Therefore, the net torque is:
net torque = force * radius = 32.144 N * 0.108 m = 3.478752 N·m
Now, we need to calculate the moment of inertia of the pulley. The moment of inertia for a disk-shaped object is given by the formula:
moment of inertia = (mass * radius^2) / 2
Plugging in the values, we get:
moment of inertia = (0.727 kg * 0.108 m^2) / 2 = 0.04415832 kg·m^2
Finally, we can calculate the angular acceleration using the formula mentioned earlier:
angular acceleration = net torque / moment of inertia
angular acceleration = 3.478752 N·m / 0.04415832 kg·m^2 ≈ 78.866 rad/s^2
So, the angular acceleration of the pulley is approximately 78.866 rad/s^2.
(c) To find how far the bucket drops in 1.65 s, we can use the formula for displacement:
displacement = initial velocity * time + (1/2) * acceleration * time^2
Since the bucket is initially at rest, the initial velocity is 0 m/s. The acceleration is the value we previously calculated as 9.8 m/s^2. Plugging in these values, we get:
displacement = 0 * 1.65 s + (1/2) * 9.8 m/s^2 * (1.65 s)^2 ≈ 8.07675 m
So, the bucket drops approximately 8.07675 meters in 1.65 s.