Forces, Density and Pressure

State the principle of moments.

By taking moments about end A, determine the distance $x$ from A to P.

distance = ______ $\text{m}$

The bottom of the cylinder is submerged in the water to depth $y$ as shown in Fig. 2.2. The beam is still attached to the cylinder but not shown.

The cylinder has mass $11\ \text{kg}$ and diameter $0.78\ \text{m}$. The beam exerts a vertical force of $1300\ \text{N}$ on the cylinder. The cylinder is in equilibrium.

Show that the upthrust acting on the cylinder is $1400\ \text{N}$.

The water has density $990\ \text{kg m}^{-3}$.

Calculate the depth $y$.

$y$ = ______ $\text{m}$

The person can stand anywhere between A and B.

On Fig. 2.3, sketch the variation of the depth of the bottom of the cylinder with the distance of the person from A, for distances between $0$ and $6.0\ \text{m}$. Numerical values are not required.

Define the moment of a force about a point.

By taking moments about point $Q$, show that $\theta$ is $25^\circ$.

On Fig. 2.2, draw a labelled scale vector triangle to represent the forces acting on the tree. The weight of the tree has been drawn to scale.

The tree exerts a pressure of $150\text{ kPa}$ on the top of the post.

Determine the surface area of the tree in contact with the post.

area = ______ $\text{m}^2$

Define the moment of a force about a pivot.

Calculate $m$.

$m$ = ______ $\text{kg}$

Object A is removed and replaced by a wire fixed to the end of the beam and to the ground, as shown in Fig. 2.2.

After the change, the beam is again horizontal and in equilibrium. The positions of B and C are unchanged.
The wire has a diameter of $1.8 \times 10^{-3}\text{ m}$ and has a strain of $1.2 \times 10^{-3}$.
The wire is not extended beyond its limit of proportionality.

Calculate the Young modulus of the wire.

Young modulus = ______ $\text{Pa}$

Object B is now moved to a new position closer to the pivot without passing it. The beam is again horizontal and in equilibrium.

State and explain the effect, if any, that this has on the strain in the wire.

Define the moment of a force.

State the principle of moments.

Calculate the component of the weight that is perpendicular to the trapdoor.

component of weight = ______ $\text{N}$

Calculate the magnitude of the force $F$.

$F$ = ______ $\text{N}$

Define the torque of a couple.

When the rod is supported in such a way that it can rotate freely within its support, the sheet hangs in equilibrium with point X vertically above the rod, as shown in Fig. 2.2.

On Fig. 2.2, draw a line to indicate the range of possible positions for the centre of gravity of the metal sheet.

Explain whether the torque applied to the rod to hold the sheet in equilibrium is clockwise or anticlockwise.

Show that the centre of gravity of the sheet has a horizontal displacement of $0.12\ \text{m}$ from the rod.

The square metal sheet has an average density of $3000\ \text{kg m}^{-3}$ and a uniform thickness of $4.0\ \text{mm}$.

Show that the side length of the sheet is $0.48\ \text{m}$.

Use the answer in (b) and the information in (c) and (d) to determine the position of the centre of gravity of the sheet. Indicate this position on Fig. 2.3 with a point labelled Y.

State the principle of moments.

Use the principle of moments to show that the upthrust $U$ exerted by the water on the cylinder is $1.4\ \text{N}$.

The density of the water is $1.0 \times 10^3\ \text{kg m}^{-3}$.

Calculate the area $A$ of the circular cross-section of the cylinder.

$A$ = ______ $\text{m}^2$

More water is gradually added to the container in (b), so that depth $h$ in Fig. 3.1 gradually increases. The length $x$ is continuously adjusted so that the system remains in equilibrium.

On Fig. 3.2, sketch the variation of $x$ with $h$. Use the space below for any working.

By using the definitions of pressure and density, show that

where $p$ is the pressure due to the liquid that is exerted on the base of the beaker and $g$ is the acceleration of free fall.

Suggest why the equation in (a) does not give the total pressure on the base of the beaker.

Fig. 4.2 shows the variation of the total pressure inside the liquid with depth $x$ below the surface.

Determine the density of the liquid.

density = ______ $\text{kg m}^{-3}$

A solid cylinder is held stationary by a wire so that the base of the cylinder is level with the surface of the liquid, as shown in Fig. 4.3.

The cylinder has length $4.0 \times 10^{-2}\ \text{m}$ and cross-sectional area $3.7 \times 10^{-4}\ \text{m}^2$. The tension in the wire is $0.53\ \text{N}$.

The cylinder is now lowered and then held stationary by the wire so that the top of the cylinder is level with the surface of the liquid.

Calculate the new tension in the wire.

tension = ______ $\text{N}$

$21\%$ of the volume of the sphere is below the surface of the water.

Calculate the density of the sphere.

density = ______ $\text{kg m}^{-3}$

Calculate the initial acceleration of the sphere.

acceleration = ______ $\text{m s}^{-2}$

The sphere accelerates upwards but remains entirely below the surface of the water.

State and explain what happens to the acceleration of the sphere as its velocity begins to increase.

State what is meant by the centre of gravity of an object.

By taking moments about P, calculate $F$.

$F$ = ______ $\text{N}$

Calculate the force exerted on the rod by the pivot.

force = ______ $\text{N}$

Use Archimedes’ principle to determine the radius $r$ of the sphere.

$r$ = ______ $\text{m}$

Calculate the magnitude and direction of the resultant moment of the forces on the rod about P.

magnitude of resultant moment = ______ $\text{N m}$
direction of resultant moment ______

Define the moment of a force about a point.

Show that the increase in the length of the spring is approximately $1.3\ \text{mm}$.

Calculate the magnitude of the moment about the pivot of the weight of the object.

moment = ______ $\text{N m}$

Use your answer in (b)(ii) to determine the increase in the tension in the spring due to the $0.472\ \text{kg}$ mass.

increase in tension = ______ $\text{N}$

Use the information in (b)(i) and your answer in (b)(iii) to determine the spring constant $k$ of the spring. Give a unit with your answer.

$k$ = ______ unit ______