State what the principle of moments says.
A hollow plastic sphere is fixed to one end of a bar. The sphere is partly under water and the bar is connected to a rigid vertical support by a pivot $P$, as shown in Fig. 3.1. The sphere has weight $0.30\,\text{N}$. The distance from $P$ to the centre of gravity of the sphere is $0.29\,\text{m}$. Take the weight of the bar to be negligible. Calculate the moment of the weight of the sphere about $P$.
The arrangement in Fig. 3.1 forms part of a device that regulates the water level in a tank. As water enters the tank, the sphere rises. This makes the bar become horizontal. Fig. 3.2 shows the system in this updated position. Here, rod $R$ applies a force that compresses a horizontal spring which controls the water supply to the tank. $R$ is located at a perpendicular distance of $0.017\,\text{m}$ above $P$. The change in the force $F$ on the spring with compression $x$ of the spring is shown in Fig. 3.3.
Use Fig. 3.3 to calculate the spring constant $k$ of the spring.
At the position shown in Fig. 3.2, the system is at rest and in equilibrium. The radius of the sphere is $0.0480\,\text{m}$ and $26.0\%$ of the sphere’s volume is below the water surface. The density of water is $1.00 \times 10^3\,\text{kg m}^{-3}$. Show that the upthrust on the sphere is $1.18\,\text{N}$.
By taking moments about $P$, determine the force exerted on the spring by the rod $R$.
Calculate the elastic potential energy $E_p$ of the compressed spring.
As the sphere moves from the position shown in Fig. 3.1 to the position shown in Fig. 3.2, the upthrust on the sphere does work. Assume that resistive forces are negligible. Explain why the work done by the upthrust is not equal to the gain in elastic potential energy of the spring.