A 4.00 bucket of water is accelerated upward by a cord of negligible mass whose breaking strength is 78.0 . If the bucket starts from rest, what is the minimum time required to raise the bucket a vertical distance of 10.0 without breaking the cord?
ma=T-mg
a=(T/m)-g= 19.5 – 9.8 = 9.7 m/s²
h=at²/2=>
t=2h/a = 2•10/9.7 = 2.06 s.
the above solution is correct except for the very last part. They did not take into account that it should have been:
t^2=(2h/a)
sqrt(t^2)=sqrt(2.06)
t=1.44
To determine the minimum time required to raise the bucket a vertical distance of 10.0 m without breaking the cord, several steps need to be followed. Let's break down the problem step by step:
Step 1: Calculate the weight of the bucket of water.
Since the weight of an object is given by the equation: weight = mass × gravity, where gravity is approximately 9.8 m/s², and the mass is given as 4.00 kg, we can calculate the weight of the bucket as follows:
weight = 4.00 kg × 9.8 m/s² = 39.2 N
Step 2: Determine the tension in the cord when the bucket is accelerating upwards.
When the bucket is accelerating upwards, the tension in the cord will be greater than the weight of the bucket. We can find the tension using the equation:
tension = weight + ma, where m is the mass of the bucket and a is the upward acceleration.
Since we want to lift the bucket, the acceleration will be positive, and the tension will be maximum when the acceleration is at its maximum. Therefore, we assume maximum acceleration to calculate the tension.
Given that the breaking strength of the cord is 78.0 N, the tension in the cord should not exceed this value.
So, maximum tension = 78.0 N
Since tension = weight + ma, we can rearrange the equation to:
ma = maximum tension - weight
Step 3: Calculate the acceleration required to raise the bucket.
Using the rearranged equation from step 2, we have:
a = (maximum tension - weight) / m
Substituting the values, we get:
a = (78.0 N - 39.2 N) / 4.00 kg
a = 19.7 m/s²
Step 4: Apply kinematic equation to find the minimum time.
Using the second kinematic equation: displacement = ut + (1/2)at², where u is the initial velocity (which is 0 since the bucket starts from rest), displacement is 10.0 m, and a is 19.7 m/s², we can rearrange the equation to:
t = sqrt(2s / a)
Substituting the values, we get:
t = sqrt(2 × 10.0 m / 19.7 m/s²)
t = 1.01 s (rounded to 3 significant figures)
Therefore, the minimum time required to raise the bucket a vertical distance of 10.0 m without breaking the cord is approximately 1.01 seconds.
To calculate the minimum time required to raise the bucket a vertical distance of 10.0 without breaking the cord, we can use Newton's second law of motion and the concept of acceleration.
First, we need to calculate the force required to lift the bucket. We know that the mass of the bucket is 4.00 kg, and we can assume the acceleration due to gravity is 9.81 m/s^2. Therefore, the weight of the bucket is given by:
Weight = mass * acceleration due to gravity
Weight = 4.00 kg * 9.81 m/s^2
Weight = 39.24 N
Since the cord has a breaking strength of 78.0 N, we need to make sure that the force applied to lift the bucket does not exceed this value.
Next, we can calculate the acceleration of the bucket. We use the equation:
Force = mass * acceleration
39.24 N = 4.00 kg * acceleration
acceleration = 9.81 m/s^2
Now, we can use the kinematic equation to find the minimum time required. The equation we can use is:
Distance = initial velocity * time + 0.5 * acceleration * time^2
Since the bucket starts from rest, the initial velocity is 0 m/s. The distance is 10.0 m, and the acceleration is 9.81 m/s^2. Plugging these values into the equation, we get:
10.0 m = 0 * t + 0.5 * 9.81 m/s^2 * t^2
10.0 m = 0 + 4.905 m/s^2 * t^2
10.0 m = 4.905 m/s^2 * t^2
Simplifying the equation, we have:
t^2 = 10.0 m / 4.905 m/s^2
t^2 = 2.04 s^2
Taking the square root of both sides, we find:
t = √(2.04 s^2)
t = 1.43 s (rounded to two decimal places)
Therefore, the minimum time required to raise the bucket a vertical distance of 10.0 m without breaking the cord is approximately 1.43 seconds.