A 5.30kg bucket of water is accelerated upward by a cord of negligible mass whose breaking strength is 76.0N .If the bucket starts from rest, what is the minimum time required to raise the bucket a vertical distance of 14.0m without breaking the cord?
Well, well, well, looks like we've got ourselves a physics problem! Don't worry, I'm here to help, but I must warn you, my answers tend to be a bit clownish. Ready for some fun?
So, we've got a bucket of water being accelerated upward by a cord. The first thing we need to figure out is the tension in the cord when the bucket is at its heaviest. We know that the force of gravity acting on the bucket is given by F = mg, where m is the mass of the bucket and g is the acceleration due to gravity.
In this case, the mass of the bucket, m, is 5.30 kg, and the acceleration due to gravity, g, is approximately 9.8 m/s^2. So, the force of gravity acting on the bucket is F = 5.30 kg * 9.8 m/s^2 = 51.94 N.
Now, the tension in the cord when the bucket is at its heaviest is equal to the force of gravity plus the force needed to accelerate the bucket upward. Let's call this tension T. So, T = F + ma, where a is the acceleration of the bucket.
We know that the acceleration of the bucket is given by a = Δv/Δt, where Δv is the change in velocity and Δt is the change in time. In this case, the bucket starts from rest, so the initial velocity is zero.
The final velocity, vf, can be found using the equation vf^2 = vi^2 + 2aΔx, where vi is the initial velocity (zero in this case), a is the acceleration, and Δx is the displacement.
The displacement Δx is given as 14.0 m, and we need to find the minimum time required, so we need to find the maximum acceleration to prevent the cord from breaking. The breaking strength of the cord is given as 76.0 N.
Now comes the fun part! To prevent the cord from breaking, the tension, T, should be less than the breaking strength of the cord, which is 76.0 N.
So, using T = F + ma, we can substitute the values we know: 76.0 N = 51.94 N + 5.30 kg * a.
Simplifying this equation, we find 24.06 N = 5.30 kg * a.
Now, solving for acceleration, we find a ≈ 4.54 m/s^2.
Finally, we can use the equation vf^2 = vi^2 + 2aΔx to find the final velocity:
vf^2 = 0 + 2 * 4.54 m/s^2 * 14.0 m. Solving for vf, we find vf ≈ 16.96 m/s.
Now, we can use the formula vf = vi + at to find the time it takes for the bucket to reach this final velocity. Since the initial velocity, vi, is zero, we can simplify the equation to t = vf / a, which gives us t ≈ 16.96 m/s / 4.54 m/s^2 ≈ 3.74 s.
So, the minimum time required to raise the bucket a vertical distance of 14.0 m without breaking the cord is approximately 3.74 seconds. But remember, this is the clown version of physics, so expect a few laughs along the way!
To find the minimum time required to raise the bucket a vertical distance of 14.0m without breaking the cord, we need to consider the forces acting on the bucket and apply Newton's second law of motion.
1. Calculate the weight of the bucket:
Weight = mass * gravity
Weight = 5.30kg * 9.8 m/s^2
Weight = 51.94 N
2. Calculate the tension in the cord when the bucket is accelerated upward:
Tension in the cord = mass * acceleration
Tension = 5.30kg * acceleration
3. Set up the equation for the tension in the cord:
Tension = Weight + mass * acceleration
5.30kg * acceleration = 51.94 N + 5.30kg * 9.8 m/s^2
Note: The acceleration will be negative since it is in the downward direction while the bucket moves upward.
4. Calculate the maximum acceleration that the bucket can have without breaking the cord:
76.0N = 51.94N + 5.30kg * acceleration
24.06N = 5.30kg * acceleration
acceleration = 24.06N / 5.30kg
acceleration = 4.54 m/s^2
5. Use the kinematic equation to find the time required to raise the bucket a vertical distance of 14.0m:
Δy = V_initial * t + (1/2) * acceleration * t^2
14.0m = 0 * t + (1/2) * 4.54 m/s^2 * t^2
Rearranging the equation, we get:
0.227t^2 = 14.0m
t^2 = 14.0m / 0.227
t^2 = 61.67
t = √61.67
t ≈ 7.85s
Therefore, the minimum time required to raise the bucket a vertical distance of 14.0m without breaking the cord is approximately 7.85 seconds.
To determine the minimum time required to raise the bucket without breaking the cord, we can use the equations of motion. First, let's identify the forces acting on the system:
1. The weight of the bucket (mg), acting downward (where m = mass of the bucket and g = acceleration due to gravity = 9.8 m/s^2).
2. The tension force in the cord (T), acting upward.
Since the bucket is being accelerated upward, there is also a net upward force acting on it. Therefore, we can write the equation of motion as:
T - mg = ma
where a is the acceleration of the bucket. Rearranging this equation, we get:
T = m(a + g)
The tension force (T) in the cord must be less than or equal to the breaking strength of the cord, which is given as 76.0N. Therefore, we can write:
m(a + g) ≤ 76.0N
Now, we need to find the acceleration (a) of the bucket. We know that the bucket starts from rest and travels a vertical distance of 14.0m. We can use the equation of motion:
v^2 = u^2 + 2as
where v is the final velocity, u is the initial velocity, s is the distance traveled, and a is the acceleration. Here, u = 0 (since the bucket starts from rest), and v is the final velocity at the top, which is also 0 (since the bucket stops momentarily before descending).
Applying this equation, we get:
0 = 0^2 + 2a(14.0)
0 = 2a(14.0)
Now, solving for a, we find:
a = 0 m/s^2
Since a = 0, we know that the bucket is not accelerating during the upward motion. Thus, the tension force (T) will only be equal to the weight of the bucket (mg).
Substituting this into the inequality, we have:
m(g + 0) ≤ 76.0N
(m)(9.8 m/s^2) ≤ 76.0N
5.30 kg × 9.8 m/s^2 ≤ 76.0N
51.94 N ≤ 76.0N
Therefore, the tension force must be less than or equal to 76.0N to avoid breaking the cord.
Since the tension force is equal to the weight, and the weight remains constant during the upward motion, the minimum time required can be found using the formula for displacement:
s = ut + (1/2)at^2
Since the initial velocity (u) is 0 and acceleration (a) is 0, the equation simplifies to:
s = ut
14.0m = 0 × t
0.0 = 14.0t
Therefore, the minimum time required is 0 seconds. The bucket can be lifted instantaneously without breaking the cord, as the tension force is always less than the breaking strength.