A 5.50 bucket of water is accelerated upward by a cord of negligible mass whose breaking strength is 77.0 . If the bucket starts from rest, what is the minimum time required to raise the bucket a vertical distance of 14.0 without breaking the cord?
To solve this problem, we can use Newton's second law of motion and the concept of tension in a vertical motion.
1. First, we need to determine the net force acting on the bucket in the upward direction. We can use Newton's second law, which states that the net force acting on an object is equal to the product of its mass and acceleration:
F_net = m * a
In this case, the mass of the bucket (m) is given as 5.50 kg, and the acceleration (a) is what we need to find.
2. The tension in the cord provides the force necessary to accelerate the bucket upward. Assuming the positive direction is upward, the net force acting on the bucket is the tension force minus the force due to gravity. The force due to gravity is given by the weight of the bucket:
F_net = T - m * g
where T is the tension and g is the acceleration due to gravity (approximately 9.8 m/s^2).
3. We want to find the minimum time required to raise the bucket, so we need to determine the minimum acceleration. For this, we consider the maximum tension the cord can handle without breaking. The breaking strength of the cord is given as 77.0 N.
T_max = 77.0 N
4. The maximum tension occurs when the acceleration is at its minimum. So, we substitute T_max into the equation:
T_max - m * g = m * a_min
5. Now we can solve for a_min:
a_min = (T_max - m * g) / m
6. We now have the minimum acceleration. To find the minimum time required, we can use one of the kinematic equations:
v = u + a * t
where v is the final velocity (which is 0 since the bucket starts from rest), u is the initial velocity (also 0), a is the acceleration, and t is time.
Rearranging the equation, we get:
t = (v - u) / a
Since v = 0 and u = 0, the equation simplifies to:
t = 0 / a
Therefore, the minimum time required to raise the bucket is 0 seconds.
In summary, the minimum time required to raise the bucket a vertical distance of 14.0 m without breaking the cord is 0 seconds because the bucket starts from rest and the minimum acceleration required is 0, which means it doesn't need any time to reach the destination.