Question

A piston weighing W = 3 kg has the form of a circular disc of radius R = 4 cm. The disc has a hole into which a thin - walled pipe of radius r = 1 cm in inserted. Initially the piston is at the bottom of the cylinder. What height will the piston rise to if m = 700 g of water is poured into the pipe?

Answer 1

\frac{5}{16\pi} \text{ m}

Answer 2

The problem involves the equilibrium of a piston under the influence of its weight and water pressure. We need to find the height the piston rises to, which is denoted as 'H' in the diagram.

1. Determine the height of water in the pipe (h) required to lift the piston: When the piston is in equilibrium, the upward force due to water pressure below it balances the downward forces due to its weight and atmospheric pressure on its top surface.

• Mass of piston (W): 3 kg
• Radius of piston (R): 4 cm = 0.04 m
• Radius of pipe (r): 1 cm = 0.01 m
• Density of water (ρ): 1000 kg/m³
• Atmospheric pressure (P₀): (acts on both top and bottom surfaces, effectively cancels out for the pressure difference that balances the weight)

Let 'h' be the height of the water column in the pipe above the level of the piston. The pressure of the water just below the piston (at the level of the piston's bottom surface) is given by: $P_{\text{water below}} = P_0 + \rho g h$

The area of the piston on which this upward pressure acts is the annular area, which is the total area of the piston minus the area of the pipe's cross-section: $A_{\text{annulus}} = \pi R^2 - \pi r^2 = \pi (R^2 - r^2)$ $A_{\text{annulus}} = \pi ((0.04)^2 - (0.01)^2) = \pi (0.0016 - 0.0001) = 0.0015\pi \, \text{m}^2$

Now, consider the force balance on the piston. Downward forces: Weight of piston (Wg) + Atmospheric force on top annular area (P₀ * A_annulus) Upward force: Water pressure force from below (P_water below * A_annulus)

For equilibrium: $Wg + P_0 A_{\text{annulus}} = (P_0 + \rho g h) A_{\text{annulus}}$ $Wg + P_0 A_{\text{annulus}} = P_0 A_{\text{annulus}} + \rho g h A_{\text{annulus}}$ The atmospheric pressure terms cancel out: $Wg = \rho g h A_{\text{annulus}}$ $h = \frac{W}{\rho A_{\text{annulus}}}$

Substitute the values: $h = \frac{3 \, \text{kg}}{1000 \, \text{kg/m}^3 \times 0.0015\pi \, \text{m}^2}$ $h = \frac{3}{1.5\pi} = \frac{2}{\pi} \, \text{m}$

2. Calculate the height the piston rises (H) using the total mass of water: The total mass of water poured (m = 700 g = 0.7 kg) is distributed in two parts:

• Water in the pipe above the piston.
• Water in the cylinder below the piston.

Let 'H' be the height the piston rises to (i.e., the height of water in the cylinder below the piston).

• Volume of water in the pipe: $V_{\text{pipe}} = \pi r^2 h$
• Volume of water in the cylinder below the piston: $V_{\text{cylinder}} = \pi R^2 H$

The total mass of water is: $m = \rho V_{\text{pipe}} + \rho V_{\text{cylinder}}$ $m = \rho (\pi r^2 h + \pi R^2 H)$

Substitute the known values: $0.7 \, \text{kg} = 1000 \, \text{kg/m}^3 \times \left( \pi (0.01 \, \text{m})^2 \left(\frac{2}{\pi} \, \text{m}\right) + \pi (0.04 \, \text{m})^2 H \right)$ $0.7 = 1000 \times \left( \pi (0.0001) \left(\frac{2}{\pi}\right) + \pi (0.0016) H \right)$ $0.7 = 1000 \times (0.0002 + 0.0016\pi H)$ $0.7 = 0.2 + 1.6\pi H$ Now, solve for H: $0.7 - 0.2 = 1.6\pi H$ $0.5 = 1.6\pi H$ $H = \frac{0.5}{1.6\pi} = \frac{5}{16\pi} \, \text{m}$

The height the piston will rise to is $\frac{5}{16\pi}$ meters.

The final answer is $\boxed{\frac{5}{16\pi} \text{ m}}$.

Explanation of the solution:

1. Pressure Balance on Piston: The weight of the piston is balanced by the upward force exerted by the water pressure from below. The effective area for this force is the annular area of the piston ($\pi R^2 - \pi r^2$). The pressure of the water below the piston is $P_0 + \rho g h$, where $h$ is the height of water in the pipe above the piston. Equating forces ($Wg = \rho g h (\pi R^2 - \pi r^2)$) yields $h = \frac{W}{\rho (\pi R^2 - \pi r^2)}$.
2. Mass Conservation: The total mass of water poured ($m$) is the sum of the mass of water in the pipe ($m_p = \rho \pi r^2 h$) and the mass of water in the cylinder below the piston ($m_c = \rho \pi R^2 H$). Using $m = m_p + m_c$, substitute the calculated $h$ and solve for $H$.

Answer:

The height the piston will rise to is $\frac{5}{16\pi}$ m.