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Chapter 3: Problem 125

A pail, \(400 \mathrm{mm}\) in diameter and \(400 \mathrm{mm}\) deep, weighs \(15 \mathrm{N}\) and contains \(200 \mathrm{mm}\) of water. The pail is swung in a vertical circle of \(1-\mathrm{m}\) radius at a speed of \(5 \mathrm{m} / \mathrm{s}\). Assume the water moves as a rigid body. At the instant when the pail is at the top of its trajectory, compute the tension in the string and the pressure on the bottom of the pail from the water.

Short Answer

Expert verified
The tension in the string and pressure at the bottom of the pail can be calculated using the centripetal force and pressure at a certain depth in a fluid.

Step by step solution

01

Identify known values

First, identify the known values from the problem. Here, they include the radius of the pail (\(r_{pail} = 0.2 \, m\)), the depth of the water (\(h = 0.2 \, m\)), the weight of the pail (\(W_{pail} = 15 \, N\)), the radius of the circular path (\(R = 1 \, m\)) and the speed (\(v = 5 \, m/s\)). The density of water is a standard value of \( \rho = 1000 \, kg/m^3 \) and the gravitational acceleration is \(g = 9.81 \, m/s^2\).
02

Calculate the Centripetal Force

Calculate the centripetal force which is responsible for the circular motion of the pail. This force can be calculated as follows \[ F_{c} = m \cdot (v^2/R) \] where \( m \) is the mass of water. Given that density \(\rho = m/V\), the mass \(m\) can be calculated as \(m = \rho \cdot V = \rho \cdot \pi \cdot (r_{pail})^2 \cdot h \)
03

Calculate the Tension in the String

Calculate the tension in the string when the pail is at the top of the trajectory. The tension is equal to the weight of the water and pail plus the centripetal force, formulated as follows: \[ T = W_{pail} + m \cdot g + F_{c} \]. This is due to the net force in circular motion being the centripetal force.
04

Calculate the Pressure on the Bottom of the Pail

Lastly, compute the pressure exerted by the water on the bottom of the pail. The pressure in a fluid at a certain depth is given by \[ P = P_{atm} + \rho \cdot g \cdot h + \rho \cdot (v^2/R) \] where \( P_{atm} \) is the atmospheric pressure (value is \(1.013 \times 10^5 \, Pa\) at sea level).

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