A wooden bucket filled with water has a mass
of 59 kg and is attached to a rope that is
wound around a cylinder with a radius of
0.071 m. A crank with a turning radius of
0.29 m is attached to the end of the cylinder.
What minimum force directed perpendicularly to the crank handle is required to raise
the bucket? The acceleration of gravity is
9.81 m/s
2
.
To find the minimum force required to raise the bucket, we need to consider the forces acting on the system.
Let's start by identifying the forces:
1. Weight of the bucket (mg): The weight of the bucket is equal to the mass of the bucket multiplied by the acceleration due to gravity. In this case, the weight (mg) is given as 59 kg multiplied by 9.81 m/s^2.
Weight of the bucket = (mass of the bucket) x (acceleration due to gravity)
Weight of the bucket = 59 kg x 9.81 m/s^2
2. Tension in the rope: The tension in the rope is the force that is needed to hold the bucket and lift it. This tension is what we need to calculate.
We know that the rope is wound around a cylinder with a radius of 0.071 m. The force applied to the rope by the crank handle is transmitted to the bucket through the rope and cylinder.
3. Force applied to the crank handle: The force we're trying to find is the minimum force required to raise the bucket. It is perpendicular to the crank handle.
Now, let's calculate the minimum force required:
The torque (force x lever arm) exerted by the force applied to the crank handle is equal to the torque exerted by the tension in the rope.
Torque exerted by the tension in the rope = Torque exerted by the force applied to the crank handle
(Tension in the rope) x (radius of the cylinder) = (Force applied to the crank handle) x (radius of the crank)
Note that the radius of the cylinder is 0.071 m, and the radius of the crank is 0.29 m.
(Tension in the rope) x (0.071 m) = (Force applied to the crank handle) x (0.29 m)
Now, we can substitute the weight of the bucket (mg) for the tension in the rope:
(mg) x (0.071 m) = (Force applied to the crank handle) x (0.29 m)
Substitute the given values:
(59 kg x 9.81 m/s^2) x (0.071 m) = (Force applied to the crank handle) x (0.29 m)
Solving for the force applied to the crank handle:
Force applied to the crank handle = [(59 kg x 9.81 m/s^2) x (0.071 m)] / (0.29 m)
After calculating this expression, we find the minimum force required to raise the bucket.
To find the minimum force required to raise the bucket, we can use the principle of torque. The torque required to lift the bucket can be calculated using the formula:
Torque = Force * Lever Arm
Here, the lever arm is the distance from the point where the force is applied (the crank handle) to the axis of rotation (the center of the cylinder).
Given:
Mass of the bucket (m) = 59 kg
Radius of the cylinder (r) = 0.071 m
Turning radius of the crank (R) = 0.29 m
Acceleration due to gravity (g) = 9.81 m/s^2
First, we need to calculate the force required to counteract the weight of the bucket. The weight (W) of the bucket can be calculated using the formula:
Weight = mass * acceleration due to gravity
W = m * g
W = 59 kg * 9.81 m/s^2
W = 577.79 N
Next, we can calculate the torque required to lift the bucket using the formula:
Torque = Force * Lever Arm
The lever arm for this scenario is the distance from the crank handle to the center of the cylinder, which is the sum of the radius of the cylinder and the turning radius of the crank.
Lever Arm = r + R
Lever Arm = 0.071 m + 0.29 m
Lever Arm = 0.361 m
Now, substituting the values into the torque formula:
Torque = Force * Lever Arm
Torque = W * Lever Arm
Torque = 577.79 N * 0.361 m
Torque = 208.71 N*m
To find the minimum force required, we need to consider the maximum torque that can be exerted by the person turning the crank. The maximum torque can be calculated as the product of the applied force (F) and the turning radius of the crank (R).
Maximum Torque = Force * Turning Radius
To find the minimum force required, we rearrange the formula as follows:
Force = Torque / Turning Radius
Force = 208.71 N*m / 0.29 m
Force = 719.00 N
Therefore, the minimum force directed perpendicularly to the crank handle required to raise the bucket is 719.00 N.
weight of bucket=50g
Now consider the crank and cylinder to be a lever.
Force*Leverarm=weigh*cylinderradius
force*.29=50g*.071
solve for force.