A cylindrical 9.16 kg pulley with a radius of
0.373 m is used to lower a 2.64 kg bucket into
a well. The bucket starts from rest and falls
for 3.94 s.
What is the linear acceleration of the
falling bucket? The acceleration of gravity
is 9.81 m/s
2
.
Answer in units of m/s
2
To find the linear acceleration of the falling bucket, we can use the following formula:
acceleration = (2 * Δy) / (Δt^2)
Where:
Δy = change in height = initial height - final height
Δt = time interval
Given:
Mass of the pulley (m1) = 9.16 kg
Mass of the bucket (m2) = 2.64 kg
Radius of the pulley (r) = 0.373 m
Time interval (Δt) = 3.94 s
Acceleration due to gravity (g) = 9.81 m/s^2
First, we need to calculate the change in height (Δy) of the bucket.
The pulley and the bucket are connected, so when the bucket falls, the pulley rotates, and the distance the bucket falls is equal to the distance the outer edge of the pulley moves.
Δy = 2πr * θ
Where:
θ = angle of rotation in radians
First, we need to find the angle of rotation (θ).
The circumference of the pulley is equal to 2πr.
Distance fallen by the bucket = Δy = 2πr * θ
Since Δy is linear, and θ is angular, we can relate them by using the formula:
Δy = r * θ
Now, we can calculate the angle of rotation (θ) using the relationship:
θ = Δy / r = Δy / (0.373 m)
Next, we can calculate the change in height (Δy) using the given masses and acceleration due to gravity.
The bucket is connected to the pulley, so the actual force acting on the bucket is:
Force on the bucket (Ftotal) = m2 * g
Using this force, we can calculate the acceleration (a) using Newton's second law:
Ftotal = m2 * a
Now, we can relate the linear acceleration (a) to the angular acceleration (α) using the following formula:
a = r * α
Finally, we can solve for the linear acceleration (a).
Let's perform the calculations step-by-step.
Step 1: Calculate the angle of rotation (θ)
θ = Δy / r = Δy / (0.373 m)
Step 2: Calculate the change in height (Δy)
Ftotal = m2 * g
Δy = Ftotal / (m1 + m2) * Δt^2
Δy = (2.64 kg * 9.81 m/s^2) / (9.16 kg + 2.64 kg) * (3.94 s)^2
Step 3: Calculate the angle of rotation (θ)
θ = Δy / r = Δy / (0.373 m)
Step 4: Calculate the linear acceleration (a)
a = r * α
Since α = θ / Δt^2, we can substitute the values:
a = (0.373 m) * (θ / Δt^2) = (0.373 m) * (Δy / (0.373 m) / (3.94 s)^2)
Finally, plug in the calculated values into the formula and solve for the linear acceleration.
a = (Δy / (3.94 s)^2)
Perform the calculation:
a = (Δy / (3.94 s)^2) = (1.68 m) / (3.94 s)^2 = 0.109 m/s^2
Therefore, the linear acceleration of the falling bucket is 0.109 m/s^2.
To find the linear acceleration of the falling bucket, we need to first calculate the tension in the rope connecting the bucket to the pulley.
The tension in the rope can be determined using Newton's second law, which states that the net force acting on an object is equal to its mass multiplied by its acceleration. In this case, the net force is the sum of the gravitational force acting on the bucket (weight) and the tension in the rope.
The gravitational force acting on the bucket is determined by its mass and the acceleration due to gravity:
gravitational force = mass * acceleration due to gravity
= 2.64 kg * 9.81 m/s^2
= 25.9344 N
As the bucket is falling, the pulley is rotating and experiencing angular acceleration. This angular acceleration is related to the linear acceleration of the bucket and the radius of the pulley.
The angular acceleration can be calculated using the formula:
angular acceleration = linear acceleration / radius
Rearranging the formula gives:
linear acceleration = angular acceleration * radius
To find the angular acceleration of the pulley, we can use the kinematic equation for rotational motion:
angular acceleration = (final angular velocity - initial angular velocity) / time
Since the pulley starts from rest, the initial angular velocity is zero.
The final angular velocity can be determined using the relationship between linear and angular motion:
linear velocity = angular velocity * radius
Rearranging the equation gives:
angular velocity = linear velocity / radius
To find the linear velocity of the bucket, we can use the formula:
linear velocity = (final position - initial position) / time
Since the bucket starts from rest, the initial position is zero. The final position can be calculated using the formula for the distance traveled by a falling object:
distance = (1/2) * acceleration * time^2
Plugging in the given values, we have:
distance = (1/2) * 9.81 m/s^2 * (3.94 s)^2
= 74.4462 m
Substituting the distance and time into the linear velocity equation, we have:
linear velocity = 74.4462 m / 3.94 s
= 18.8899 m/s
Finally, substituting the linear velocity and radius into the equation for angular velocity, we have:
angular velocity = 18.8899 m/s / 0.373 m
= 50.6504 rad/s
Using this value for the final angular velocity and the given time, we can calculate the angular acceleration:
angular acceleration = (50.6504 rad/s - 0 rad/s) / 3.94 s
= 12.866 rad/s^2
Finally, substituting the angular acceleration and pulley radius into the equation for linear acceleration, we have:
linear acceleration = 12.866 rad/s^2 * 0.373 m
= 4.802 m/s^2
Therefore, the linear acceleration of the falling bucket is approximately 4.802 m/s^2.