A 5.40 kg bucket of water is accelerated upward by a cord of negligible mass whose breaking strength is 70.0 N .If the bucket starts from rest, what is the minimum time required to raise the bucket a vertical distance of 15.0 m without breaking the cord?
To solve this problem, we will use Newton's second law and the work-energy principle.
First, let's find the tension in the cord when the bucket is being accelerated upward.
From Newton's second law, we have
Tension - weight = mass * acceleration
Tension - m * g = m * a (Equation 1)
where Tension is the tension in the cord, weight is the weight of the bucket, m is the mass of the bucket, g is the acceleration due to gravity, and a is the acceleration of the bucket.
Since the bucket is being accelerated upward, the acceleration and the tension in the cord are in the same direction. Therefore, the tension in the cord will be greater than the weight of the bucket.
The breaking strength of the cord is given as 70.0 N. Thus, the tension in the cord must be less than or equal to 70.0 N for the cord not to break.
Therefore, we can rewrite Equation 1 as:
Tension ≤ m * g + m * a (Equation 2)
Next, let's find the maximum acceleration the bucket can have without breaking the cord.
When the tension in the cord is equal to the breaking strength (70.0 N), we have:
70.0 N = 5.40 kg * g + 5.40 kg * a (Equation 3)
Now, let's find the minimum time required to raise the bucket a vertical distance of 15.0 m without breaking the cord.
Using the work-energy principle, we have:
Work done on the bucket = change in potential energy of the bucket
Work done on the bucket = force * distance
Force = tension in cord
Distance = vertical distance raised (15.0 m)
Change in potential energy = m * g * h
where h is the vertical distance raised (15.0 m).
Therefore, we can write:
tension * 15.0 m = 5.40 kg * g * 15.0 m
Tension = 5.40 kg * g (Equation 4)
Using Equation 4, we can rewrite Equation 3 as:
5.40 kg * g = 5.40 kg * g + 5.40 kg * a
0 = 5.40 kg * a (Equation 5)
Since Equation 5 implies that a = 0, it means that the bucket must be moving at a constant velocity. In other words, it will take an infinite amount of time to raise the bucket 15.0 m without breaking the cord.
Therefore, it is not possible to raise the bucket 15.0 m without breaking the cord if the maximum tension the cord can withstand is 70.0 N.