Solve SHM Leaky Bucket: dT/dt Calculus Solution

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In summary, we were discussing a bucket with a leak in it and its vertical simple harmonic motion. We were asked to find the rate at which the period changes with time, given the rate of water leaking from the bucket, the mass of the bucket and total water, starting amplitude, and the spring constant. After attempting the solution, it was pointed out that the units for the rate of water leaking needed to be converted from g/s to kg/s.
  • #1
PsychonautQQ
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Homework Statement


There is a bucket with a leak in it in verticle SHM. What is the rate that the period changes with time?
dm/dt = 2 g/s
bucket = 2 kg
total water = 10kg
starting amplitude = 3cm
k=125 N/m

Homework Equations


T=2pi*sqrt(m/k)

The Attempt at a Solution



dT/dt = 2pi * sqrt((dm/dt)/k)
dT/dt = 2pi * (dm/dt)^1/2 / k^(1/2)
dT/dt = 2pi * (m/2)^-1/2 / k^1/2
dT/dt = 2pi * (m2k)^-1/2

my calculus isn't the sharpest so I probably screwed up taking the derivative?
 
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  • #2
Yeah, I think you did. Take a look at the first line again. You clearly know ##\sqrt{A}=A^{\frac{1}{2}}##, so perhaps make that conversion for ##T## before taking the derivative. Also, you will need the chain rule for this problem.
 
  • #3
PsychonautQQ said:

dm/dt = 2 g/s


What does this mean? Units have to be kg/s, not g/s.
 

FAQ: Solve SHM Leaky Bucket: dT/dt Calculus Solution

1. What is SHM and how does it relate to the Leaky Bucket problem?

SHM stands for Simple Harmonic Motion, which is a type of periodic motion where the restoring force is directly proportional to the displacement from equilibrium. In the Leaky Bucket problem, SHM is used to model the motion of the water as it leaks out of the bucket.

2. How does calculus help to solve the Leaky Bucket problem?

Calculus is used to find the rate of change of the water level in the bucket, which is essential for solving the Leaky Bucket problem. Specifically, we use the derivative of the water level function to find the rate of change over time, which helps us determine how quickly the water level is decreasing due to the leak.

3. What is the equation used to model the water level in the Leaky Bucket problem?

The equation used is dT/dt = -k(T-T0), where T is the water level, t is time, k is the leak rate constant, and T0 is the initial water level. This equation is derived using the principles of SHM and calculus.

4. How can we use the solution to the Leaky Bucket problem in real life?

The solution can be applied to real-life situations where a container is leaking a substance, such as a water tank or gas tank. By understanding the rate at which the substance is leaking, we can estimate how long it will take for the container to become empty and take necessary actions to prevent it.

5. Are there any limitations to the SHM Leaky Bucket problem and its solution?

Yes, there are limitations as this problem only models a simplified version of real-life scenarios. It assumes that the leak rate is constant, the container is a perfect cylinder with no other forces acting on it, and there is no evaporation or other external factors affecting the water level. In reality, these factors may vary and affect the accuracy of the solution.

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